- Heyting arithmetic
In

mathematical logic ,**Heyting arithmetic**is an axiomatization of arithmetic in accordance with the philosophy ofintuitionism . It is named afterArend Heyting , who first proposed it.Heyting arithmetic adopts the axioms of

Peano arithmetic , but usesintuitionistic logic as its rules of inference. In particular, thelaw 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 $forall\; x,y\; in\; mathbb\{N\}\; :\; x\; =\; y\; vee\; x\; e\; y$ 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 anyquantifier -free formula "p", $forall\; x,y,z,...\; in\; mathbb\{N\}\; :\; p\; vee\; eg\; p$ is a theorem (where "x","y","z"... are thefree variables in "p").Heyting arithmetic should not be confused with

Heyting algebra s, which are the intuitionistic analogue ofBoolean algebra s.**External links***

Stanford Encyclopedia of Philosophy : " [*http://plato.stanford.edu/entries/logic-intuitionistic/#IntNumTheHeyAri Intuitionistic Number Theory*] " byJoan Moschovakis .

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