Heyting arithmetic

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In mathematical logic, Heyting arithmetic is an axiomatization of arithmetic in accordance with the philosophy of intuitionism. It is named after Arend Heyting, who first proposed it.

Heyting arithmetic adopts the axioms of Peano arithmetic, but uses intuitionistic logic as its rules of inference. In particular, the law of the excluded middle does not hold in general, though the induction axiom can be used to prove many specific cases. For instance, one can prove that $\forall x,y \in \mathbb{N} : x = y \vee x \ne y$ is a theorem (any two natural numbers are either equal to each other, or not equal to each other). In fact, since "=" is the only predicate symbol in Heyting arithmetic, it then follows that, for any quantifier-free formula p, $\forall x,y,z,... \in \mathbb{N} : p \vee \neg p$ is a theorem (where x,y,z... are the free variables in p).