Kleene’s (Second) Recursion Theorem
Statement
Let $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} Q : \bb N^2 \to \bb N$ be partial computable. Then exists an index $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} e$ so that $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \varphi_e(n) = Q(e, n)$ on all $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} n$. That is, the program $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \varphi_e$ is able to “see itself”
Proof
We prove this using Rogers’ fixpoint theorem. Take some $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} Q$ partial computable, and let $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} f(e) = \t{the index of } (n \mapsto Q(e, n))$ then by Rogers’, we know that exists some $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} e$ for which $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \varphi_e = \varphi_{f(e)}$ Which is to say that $% shorthands \newcommand{\cl}{ \mathcal{#1} } \newcommand{\sc}{ \mathscr{#1} } \newcommand{\bb}{ \mathbb{#1} } \newcommand{\fk}{ \mathfrak{#1} } \renewcommand{\bf}{ \mathbf{#1} } \renewcommand{\sf}{ \mathsf{#1} } % category names \newcommand{\cat}{{ \sf{#1} }} % more shorthands \newcommand{\floor}{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}{ { \lceil {#1} \rceil } } \newcommand{\ol}{ \overline{#1} } \newcommand{\t}{ \text{#1} } \newcommand{\norm}{ { \lvert {#1} \rvert } } % norm/magnitude \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}{ \langle {#1} \rangle } % tuples % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } % represents an anonymous parameter % eg. f(\apar) usually denotes the function x \mapsto f(x) \newcommand{\apar}{ {-} } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" \newcommand{\pre}{{ \small {#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } % good enough definition of yoneda \newcommand{\yo}{よ} \varphi_e = n \mapsto Q(e, n)$