Model Thy HW #2
Maynard (legally, Eli Maynard)
4.5.1
(a) Take M=(X,<) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \cl M = (X, <) a dense linear order. Show that any two elements of X % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} X realize the same 1-type over % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \varnothing. Answer revised since previous submission Take x1,x2X % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} x_1, x_2 \in X. WTS that tpM(x1/)=tpM(x2/) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \t{tp}^\cl M(x_1 / \varnothing) = \t{tp}^\cl M(x_2 / \varnothing). In other words, WTS that x1 % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} x_1 and x2 % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} x_2 satisfy the same 1-ary L % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \cl L_\varnothing-formulas in M % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \cl M. Take some 1-ary L % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \cl L-formula ρ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \rho for which Mρ(x1) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \cl M \vDash \rho(x_1). Since DLO admits quantifier elimination, and since our language has no function- or constant- symbols, then WLOG ρ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \rho is a boolean combination of the two atomic formulas x1<x1 % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} x_1 < x_1 and x1=x1 % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} x_1 = x_1. The evaluation of such a sentence does not depend on the choice of x1 % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} x_1, and so Mρ(x1)    Mρ(x2) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \cl M \vDash \rho(x_1) \iff \cl M \vDash \rho(x_2), and so the types in question are the same.
(b) Show that for a,bQ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a, b \in \bb Q, tpQ(a/N)=tpQ(b/N) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \t{tp}^\bb Q(a / \bb N) = \t{tp}^\bb Q(b / \bb N) exactly when Q % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \bb Q admits an automorphism fixing N % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \bb N and taking a % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a to b % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} b. Answer NOT revised since previous submission I’m assuming our language is L={<} % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \cl L = \{ < \}? (⇒) Assume a % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a and b % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} b have the same type over N % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \bb N. If either a % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a or b % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} b satisfy the proposition ϕ(x):=(x=n) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi(x) := (x = n) for some nN % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} n \in \bb N, then we know the other does too and hence a=b=n % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a = b = n. In this case, the identity function is a satisfactory automorphism. Otherwise, one of a % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a or b % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} b does not satisfy the given ϕ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi for any n % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} n; WLOG assume it’s a % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a. Then a % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a is non-integral, so for some nN % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} n \in \bb N, a % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a satisfies ψ(x)=n<x<n+1 % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \psi(x) = n < x < n+1. Since a % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a and b % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} b share a type, then also MNψ(b) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \cl M_\bb N \vDash \psi(b). This means that both a % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a and b % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} b lie in some open range (n,n+1) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} (n, n+1). Let σ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \sigma be an automorphism over ((n,n+1),<) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} ((n,n+1), <) taking a % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a to b % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} b (one exists by thinking of ((n,n+1),<) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} ((n,n+1),<) as a smaller copy of (Q,<) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} (\bb Q, <)). Let σ~ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \tilde \sigma be σ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \sigma lifted to (Q,<) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} (\bb Q, <) by acting as the identity everywhere else. Then σ~ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \tilde \sigma is a satisfactory automorphism. (⇐) Call such an automorphism σ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \sigma, and take ϕtpQ(a/N) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi \in \t{tp}^\bb Q(a / \bb N). Then ϕ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi is a 1-ary LN % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \cl L_\bb N-formula satisfied by a % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a in QN % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \bb Q_\bb N where QN % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \bb Q_\bb N denotes the natural extension of Q % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \bb Q into LN % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \cl L_\bb N-formulas. Construct a new formula by replacing each constant symbol nN % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} n \in \bb N ocurring in ϕ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi with a variable symbol vn % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} v_n; call the result ϕ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi' and let n % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \ol n be the tuple of all such n % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} n. Then ϕ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi' is an L % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \cl L-sentence of arity greater than 1 such that Qϕ(a,n) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \bb Q \vDash \phi'(a, \ol n). Since σ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \sigma is an automorphism of Q % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \bb Q then also Qϕ(σ(a),σ(n)) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \bb Q \vDash \phi'(\sigma(a), \sigma(\ol n)); since σ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \sigma fixes all nN % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} n \in \bb N and takes a % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a to b % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} b then by substitution Qϕ(b,n) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \bb Q \vDash \phi'(b, \ol n). Now replacing each variable vn % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} v_n in ϕ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi' with n % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} n we recover the LN % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \cl L_\bb N-formula ϕ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi and conclude that QNϕ(b) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \bb Q_\bb N \vDash \phi(b) and hence ϕtpQ(b/N) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi \in \t{tp}^\bb Q(b / \bb N).
(c) Take A={1n1:nN}{2+n1:nN} % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} A = \{1 - n^{-1} : n \in \bb N\} \cup \{ 2 + n^{-1} : n \in \bb N \}. Show that 1 and 2 realize the same type over A % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} A, but there is no automorphism of Q % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \bb Q fixing A % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} A and sending 1 to 2. Answer revised since previous submission (both parts) (pt1) Take ϕtp(1/A) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi \in \t{tp}(1/A). Note that L % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \cl L has no constant or function symbols, so ϕ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi has as terms only variables and constant symols from A % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} A. Let a % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \ol a be those constant symbols and ϕ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi' be ϕ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi with a % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \ol a replaced by free variables, so that QAϕ(x)    Qϕ(x,a) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \bb Q_A \vDash \phi(x) \iff \bb Q \vDash \phi'(x, \ol a) Then Qϕ(1,a) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \bb Q \vDash \phi'(1, \ol a). But DLO admits quantifier elimination, so WLOG ϕ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi' is quantifier-free and thus a boolean combination over atomic sentences of the form < % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \square < \square or = % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \square = \square where each % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \square is populated by either the free variable x % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} x or a free variable aa % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a \in \ol a. Any such atomic sentence produces the same result when evaluated on x=1 % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} x = 1 versus x=2 % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} x = 2. Inequalities x<ai % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} x < a_i are true exactly when ai>2 % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a_i > 2, and inequalities ai<x % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a_i < x exactly when ai<1 % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a_i < 1; equalities x=ai % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} x = a_i and ai=x % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a_i = x are always false; inequalities ai<aj % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a_i < a_j and equalities ai=aj % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a_i = a_j and x=x % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} x = x likewise do not depend on the choice of x % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} x. As such, Q % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \bb Q observes the same evaluation for ϕ(1,a) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi'(1, \ol a) and ϕ(2,a) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi'(2, \ol a). Hence by construction of ϕ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi, uh, QA % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \bb Q_A observes the same evaluation for ϕ(1) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi(1) and ϕ(2) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi(2). Thus the types in question are equal. (pt2) Let σ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \sigma be an automorphism of Q % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \bb Q sending 1 to 2. Let 1<x<2 % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} 1 < x < 2. Then σ(1)<σ(x)<σ(2) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \sigma(1) < \sigma(x) < \sigma(2), so 2<σ(x) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} 2 < \sigma(x) by replacement. Let 2<aA<σ(2) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} 2 < \underset{\in A}a < \sigma(2); such an a % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a must exist because elements of A % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} A get arbitrarily close to 2. Then σ1(2)<σ1(a)<σ1(σ(2)) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \sigma^{-1}(2) < \sigma^{-1}(a) < \sigma^{-1}(\sigma(2)), so by replacement 1<σ1(a)<2 % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} 1 < \sigma^{-1}(a) < 2 so σ1(a)(1,2) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \sigma^{-1}(a) \in (1, 2). But A(1,2)= % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} A \cap (1, 2) = \varnothing, so σ1(a)A % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \sigma^{-1}(a) \notin A, so σ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \sigma does not fix A % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} A. Contradiction!
4.5.3
Show that if a,bMn % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \ol a, \ol b \in \cl M^n and tpM(a/A)=tpM(b/A) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \t{tp}^\cl M(\ol a / A) = \t{tp}^\cl M(\ol b / A) then tpM(a/dcl(A))=tpM(b/dcl(A)) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \t{tp}^\cl M(\ol a / \t{dcl}(A)) = \t{tp}^\cl M(\ol b / \t{dcl}(A)) where dcl % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \t{dcl} denotes the definable closure Answer revised since previous submission Assume a % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \ol a and b % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \ol b have the same type over A % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} A. Take ϕ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi in the type of a % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \ol a over dcl(A) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \t{dcl}(A). Then we know ϕ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi is an n % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} n-ary Ldcl(A) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \cl L_{\t{dcl}(A)}-sentence true of a % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \ol a in Mdcl(A) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \cl M_{\t{dcl}(A)}. Constant symbols in ϕ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi are either from L % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \cl L, are from A % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} A, or are from dcl(A)A % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \t{dcl}(A) \setminus A. We can remove constants of the third kind by using the fact that they are definable from A % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} A. In doing so, we obtain an n % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} n-ary LA % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \cl L_A-formula true of a % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \ol a in MA % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \cl M_A. (For instance, on L={+} % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \cl L = \{ + \} and M=(Z;+) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \cl M = (\bb Z; +) the Ldcl({0,1}) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \cl L_{\t{dcl}(\{0, 1\})}-sentence θ(x):x=3 % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \theta(x) : x = -3becomes the L{0,1} % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \cl L_{\{0,1\}}-sentence θ(x):(s)(s+(1+1+1)=0x=s) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \theta(x) : (\forall s)(s + (1 + 1 + 1) = 0 \to x = s).) Apply this process to ϕ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi and call the result ϕ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi'. Since ϕ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi' is true of a % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \ol a in MA % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \cl M_A then ϕ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi' is in the type of a % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \ol a over A % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} A, and so it is in the type of b % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \ol b over A % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} A (since they are equal). Now reversing the same process as before, re-introduce constants in dcl(A) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \t{dcl}(A) to conclude that ϕ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi is true of b % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \ol b in Mdcl(A) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \cl M_{\t{dcl}(A)}; done.
4.5.10
Suppose AB % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} A \subseteq B and θ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \theta is a formula with parameters aA<ω % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \ol a \in A^{<\omega} and θ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \theta isolates tpM(a/B) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \t{tp}^\cl M(\ol a/B). Show that θ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \theta isolates tpM(a/A) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \t{tp}^\cl M(\ol a/A). Answer added since previous submission Since θ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \theta isolates tpM(a/B) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \t{tp}^\cl M(\ol a / B) then for each LB % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \cl L_B-formula ψ=ψ(v) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \psi = \psi(\ol v) for which MBψ(a) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \cl M_B \vDash \psi(\ol a) we know MB(v:θ(v)ψ(v)) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \cl M_B \vDash (\forall \ol v : \theta(\ol v) \to \psi(\ol v)). This universal applies in particular to formulas from tpM(a/A) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \t{tp}^\cl M(\ol a/A), considering LA % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \cl L_A-formulas as a subset of LB % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \cl L_B-formulas. Hence θ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \theta isolates tpM(a/A) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \t{tp}^\cl M(\ol a/A).
4.5.13
Take Δ % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \Delta a set of L % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \cl L-formulas closed under conjunction, disjunction, and negation; define SnΔ(T):={ΣΔ:ΣT satisfiable and containing either ϕ or ¬ϕ for each ϕΔ} % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} S_n^\Delta(T) := \{ \Sigma \subseteq \Delta : \Sigma \cup T \t{ satisfiable and containing either } \phi \t{ or } \neg \phi \t{ for each } \phi \in \Delta \} Answer not completed, hehe
(a) Show that when pSnΔ(T) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} p \in S^\Delta_n(T) exists a superset qSn(T) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} q \in S_n(T)
(b) Suppose that for each n % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} n and pSnΔ(T) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} p \in S^\Delta_n(T) exists a unique supsetset qSn(T) % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} q \in S_n(T). Show that T % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} T has quantifier elimination.