**Theorema Posner–Robinson** est theorema theoriae facultatis calculandi de gradibus Turing.

Expositio
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**Theorema.** Sit {\displaystyle B} copia numerorum naturalium, quae calculari non potest. (Id est {\displaystyle B\not \leq \_{T}\mathbf {0} }.) Tum est copia {\displaystyle G} ut
{\displaystyle G\oplus B\equiv \_{T}G^{\prime }}.

Demonstratio
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Sit {\displaystyle B}. Volumus facere series {\displaystyle ((\Phi \_{p},{\bar {X}}\_{p}))\_{p\in \omega }}, quorum:

* {\displaystyle \Phi \_{p}} est copia finita rerum sicut {\displaystyle (a,b,X)}, in qua {\displaystyle a,b\in \omega }, {\displaystyle X\in 2^{<\omega }}. {\displaystyle \Phi \_{p}(X)} est functio ab {\displaystyle 2^{\omega }} in {\displaystyle 2^{\omega }}, ut {\displaystyle \Phi \_{p}(X)(a)=b} si et tantum si {\displaystyle (a,b,X)\in \Phi \_{p}},
* {\displaystyle {\bar {X}}\_{p}} est copia finita rerum {\displaystyle X\in 2^{\omega }},

in qua serie, si {\displaystyle q>p}:

* (a) Cuique {\displaystyle (x\_{q},y\_{q},\eta )\in \Phi \_{q}\backslash \Phi \_{p}} et {\displaystyle (x\_{p},y\_{p},\sigma )\in \Phi \_{p}}, {\displaystyle \vert \eta \vert >\vert \sigma \vert },
* (b) {\displaystyle \Phi \_{p}\subseteq \Phi \_{q}} et {\displaystyle {\bar {X}}\_{p}\subseteq {\bar {X}}\_{q}},
* (c) Cuique {\displaystyle X\in {\bar {X}}\_{p}}, {\displaystyle \Phi \_{q}(X)=\Phi \_{p}(X)}.

ita ut, si {\displaystyle \Phi :=\bigcup \_{p\in \omega }\Phi \_{p}}, tum {\displaystyle \Phi (B)=\Phi ^{\prime }}. Sit {\displaystyle \Phi \_{0}={\bar {X}}\_{0}=\varnothing }. Si habemus {\displaystyle (\Phi \_{p},{\bar {X}}\_{p})}, sic agamus ut {\displaystyle (\Phi \_{p+1},{\bar {X}}\_{p+1})} faciamus:

1. Sit {\displaystyle \exists n\,\theta \_{p}(n,Z\vert n)} formula numero {\displaystyle p}. Si nancisci possumus copiam {\displaystyle \Psi \supseteq \Phi } quae (a) satisfaciat, ut cuique {\displaystyle X\in {\bar {X}}\_{p}\cup \{B\}}, {\displaystyle \Psi (X)=\Phi (X)}, tum sit {\displaystyle \Phi \_{p+1}:=\Psi \cup \{(p,1,B)\}} et {\displaystyle {\bar {X}}\_{p+1}:={\bar {X}}\_{p}}.
2. Sin nancisci non possumus {\displaystyle \Psi }, tum sciamus cuique copiae {\displaystyle \Psi \subseteq \Phi } quae (a) satisfaciat esse {\displaystyle X\in {\bar {X}}\_{p}\cup \{B\}} ut {\displaystyle \Psi (X)\neq \Phi (X)}. Sit igitur collectio {\displaystyle {\mathcal {Z}}} ut:

{\displaystyle {\bar {Z}}\in {\mathcal {Z}}\iff {\text{Sit }}\Psi \supseteq \Phi {\text{ quae (a) satisfaciat, tum si }}\exists n<\vert \Psi \vert \,\theta \_{p}(n,\Psi \vert n){\text{, tum }}\exists (a,b,X)\in \Psi \backslash \Phi {\text{, ut }}X\in {\bar {Z}}}
Sed scilicet {\displaystyle {\bar {X}}\_{p}\cup \{B\}\in {\mathcal {Z}}}, ergo {\displaystyle {\mathcal {Z}}\neq \varnothing }. Lemma habemus:
**Lemma.** (*de conis vitandis*) Sit collectio non vacua {\displaystyle {\mathcal {A}}} ut {\displaystyle X\in {\mathcal {A}}\iff \forall n\,\theta (n,X\vert n)}, in quo {\displaystyle \theta \in \Delta \_{0}} (vide pagina de hierarchia arithmetica). Sit {\displaystyle D\not \leq \_{T}\mathbf {0} }. Tum est {\displaystyle G\in {\mathcal {A}}} ut {\displaystyle G\not \geq \_{T}D}.
Ergo, sit {\displaystyle {\bar {D}}=B}, tum est {\displaystyle {\bar {Z}}\in {\mathcal {Z}}} ut {\displaystyle {\bar {Z}}\not \geq \_{T}B}, id est, {\displaystyle B\not \in {\bar {Z}}}. Igitur sit {\displaystyle \Phi \_{p+1}=\Phi \_{p}\cup \{(p,0,B)\}} et {\displaystyle {\bar {X}}\_{p+1}:={\bar {X}}\_{p}\cup {\bar {Z}}}.
Nunc tota serie facta habemus {\displaystyle \Phi } ut {\displaystyle \Phi (B)=\Phi ^{\prime }}. Ergo {\displaystyle \Phi \oplus B\equiv \_{T}\Phi ^{\prime }}, quod erat demonstrandum.

Significatio
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Hoc theorema naturam omnium copiarum quae calculari non possunt patefacit. Si {\displaystyle B} est copia quae calculari non potest, tum nancisci possumus copia {\displaystyle G}, ut differentia inter {\displaystyle G} et {\displaystyle G^{\prime }} sit {\displaystyle B}.