Church's Thesis (constructive Mathematics)

Church's Thesis (constructive Mathematics)

In constructive mathematics, Church's thesis (CT) is an axiom which states that all total functions are computable. The axiom takes its name from the Church–Turing thesis, which states that every effectively calculable function is computable function, but the constructivist version is much stronger, claiming that every function is computable.

The axiom CT is incompatible with classical logic in sufficiently strong systems. For example, Heyting arithmetic (HA) with CT as an addition axiom is able to disprove some instances of the law of the excluded middle. However, Heyting arithmetic is equiconsistent with Peano arithmetic (PA) as well as with Heyting arithmetic plus Church's thesis. That is, adding either the law of the excluded middle or Church's thesis does not make Heyting arithmetic inconsistent, but adding both does.