Computably Inseparable

Definition

Let

%% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } A, B \subseteq \bb N

be disjoint sets. We say that

%% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } A

and

%% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } B

are computably inseparable (or c.i.) if for every computable

%% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } X \subseteq \bb N

, it is not the case that

%% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } X

separates

%% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } A

and

%% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } B

; ie, it is not the case that

%% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } X \supseteq A

and

%% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } X \cap B = \varnothing

Properties

(From GGCC) Let

%% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } A = \{ n : \varphi_n(n) = 0 \}

and

%% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } B = \{ n : \varphi_n(n) = 1 \}

, then

%% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } A

and

%% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } B

are both c.e. Also, they are c.i., as follows. Assume for contradiction existence of computable

%% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } C

with

%% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } C \supsetneq A

and

%% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } C \cap B = \varnothing

. Since

%% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } C

is computable then exists

%% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } c

so that the indicator function

%% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } 1_C

is equal to

%% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \varphi _c

. Then:

%% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\impliesX}[1] { \underset {#1} {\;\;\;\; \implies\;\;\;\; } } c \in C \impliesX{\t{\text{defn} \text{of} } 1_c} 1_c(c) = 1 \impliesX{\t{\text{defn} \text{of} } c} \varphi_c(c) = 1 \impliesX{\t{\text{defn} \text{of} } B} c \in B \impliesX{C \cap B = \varnothing} c \notin C

and

%% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\impliesX}[1] { \underset {#1} {\;\;\;\; \implies\;\;\;\; } } c \notin C \impliesX{\t{\text{defn} \text{of} } 1_c} 1_c(c) = 0 \impliesX{\t{\text{defn} \text{of} } c} \varphi_c(c) = 0 \impliesX{\t{\text{defn} \text{of} } A} c \in A \impliesX{A \subseteq C} c \in C

And

%% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\impliesX}[1] { \underset {#1} {\;\;\;\; \implies\;\;\;\; } } c \in C \iff c \notin C

is a contradiction. Hence the assumption that

%% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\impliesX}[1] { \underset {#1} {\;\;\;\; \implies\;\;\;\; } } C

exists must be false.

Any pair of co-ce. sets can be computably separated