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Realizability is a collection of methods in proof theory used to study constructive proofs and extract additional information from them.
17
votes
Accepted
Kleene realizability in Peano arithmetic
$\let\T\mathrm\def\kr{\mathrel{\mathbf r}}$Let me first answer a slightly modified question:
Proposition: For any sentence $\phi$, the following are equivalent:
There exists $n\in\mathbb N$ such tha …
7
votes
Accepted
What is the theory of statements with a provably *bounded* realizer (according to PA)?
$\let\T\mathrm\def\kr{\mathrel{\mathbf r}}\let\LOR\bigvee$The same argument as in my linked answer shows that
$$T_2=\T{HA+ECT_0+MP+SWLEM},$$
where
$$\T{SWLEM}=\{\neg\phi\lor\neg\neg\phi:\text{$\phi$ s …