Set Theory HW #11
p200 ex23
Take $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} A$ a set and let $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \alpha$ be the set of ordinals that inject into $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} A$. Show that $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \alpha$ is the least cardinal greater than $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \card(A)$.
$% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \alpha$ is a cardinal:
$% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \alpha$ is transitive — Take $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a \in b \in \alpha$. Then $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} b$ is an ordinal which injects into $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} A$. Since $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} b$ is an ordinal, then it is transitive so $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a \subseteq b$, so also $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a$ injects into $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} A$. Also, since $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} b$ is an ordinal and $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a \in b$ then $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a$ is an ordinal. Thus $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a$ is an ordinal that injects into $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} A$, so $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a \in \alpha$.
$% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \in_\alpha$ is linear — Given $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a, b \in \alpha$ distinct, both $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a$ and $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} b$ are ordinals, so exactly one of $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a \in b$ and $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} b \in a$ hold; done.
If $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a \in \alpha$ then $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a \ncong \alpha$ — Note that $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \neg(A \surjects \alpha)$, as follows. Assume $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} f : A \to \alpha$, and let $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \beta = f \llbracket A \rrbracket$. Then $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \beta$ is a set of ordinals injecting into $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} A$. If $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \beta$ has no $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \in$-maximum, then let $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} m = \bigcup \beta$; since $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} m$ is a union of ordinals injecting into $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} A$ then also $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} m$ injects into $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} A$ (the truth of this has been divined directly from the Platonic Realm and requires no proof). Also $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} m$ is a maximum of $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \beta \cup \{m\}$; since $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \beta$ has no maximum then it must be that $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} m \notin \beta$. Hence $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} m \in \alpha \setminus \beta$, so $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \beta \neq \alpha$. If, on the other hand, $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \beta$ has an $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \in$-maximum, then let $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} m$ be $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \max(\beta) + 1$; again $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} m \in \alpha \setminus \beta$ and so $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \beta \neq \alpha$. Either way $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \beta \neq \alpha$ and so $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} f$ is not a surjection. Now assume for contradiction that exists an $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a \in \alpha$ with $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a \cong \alpha$. Since $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a \injects A$ and $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} a \cong \alpha$ then $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \alpha \injects A$ and so $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} A \surjects \alpha$; contradiction.
$% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \alpha > \card(A)$: We saw above that that $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \neg(A \surjects \alpha)$, so $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \card(A) \ngeq \alpha$. Since $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \in$ is linear over cardinals, then it must be that $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \alpha > \card(A)$.
$% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \alpha$ is least: Take $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \beta > \card(A)$. If an ordinal $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \gamma$ abides by $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \card(\gamma) < \card(A)$ then $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \gamma \in \card(A) < \beta$ so $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \gamma \in \beta$. Hence all ordinals $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \gamma$ injecting into $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} A$ must lie in $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \beta$, so $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \alpha \subseteq \beta$. Thus $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \alpha$ is minimal, and by linearity of $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \subseteq$ on cardinals $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \alpha$ is least.
p200 ex25
Take an arity-1 formula $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi$. Assume that for every ordinal $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \alpha$ we know $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} (\forall x \in \alpha)( \phi(x) ) \to \phi(\alpha)$. Show then that $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi$ holds on every ordinal. Take $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi$ and an ordinal $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \beta$. Let $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} A = \{ a \in \beta : \phi(a) \}$. Take $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} t \in \beta$ with $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \t{seg}(t) \subseteq A$; then $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} t$ is an ordinal abiding by $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} (\forall x \in t)(\phi(x))$ and so by assumption we get $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi(t)$. Hence when $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \t{seg}(t) \subseteq A$ we have $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} t \in A$, so by the transfinite induction principle we have that $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} A = \beta$. Therefore $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small {#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \phi$ holds over all of $% shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} \beta$.