heyting arithmetic造句

• Arend Heyting would introduce Heyting algebra and Heyting arithmetic.
• Stephen Cole Kleene ( 1945 ) proved that Heyting arithmetic has the disjunction property and the existence property.
• However, Heyting arithmetic is equiconsistent with Peano arithmetic ( PA ) as well as with Heyting arithmetic plus Church's thesis.
• However, Heyting arithmetic is equiconsistent with Peano arithmetic ( PA ) as well as with Heyting arithmetic plus Church's thesis.
• Via the G鰀el Gentzen negative translation, the consistency of classical Peano arithmetic had already been reduced to the consistency of intuitionistic Heyting arithmetic.
• It deals with Heyting arithmetic, where the domain of quantification is the natural numbers and the primitive propositions are of the form x = y.
• Is there any theory of arithmetic within the framework of linear logic, analogous to how Heyting arithmetic is the arithmetic of intuitionistic or constructive logic?
• G鰀el's motivation for developing the dialectica interpretation was to obtain a relative consistency proof for Heyting arithmetic ( and hence for Peano arithmetic ).
• For instance, in Heyting arithmetic, one can prove that for any proposition " p " that " does not contain infinite collections.
• For example, Heyting arithmetic ( HA ) with CT as an addition axiom is able to disprove some instances of the law of the excluded middle.
• That is, adding either the law of the excluded middle or Church's thesis does not make Heyting arithmetic inconsistent, but adding both does.
• Formulas and proofs in classical arithmetic can also be given a dialectica interpretation via an initial embedding into Heyting arithmetic followed the dialectica interpretation of Heyting arithmetic.
• Formulas and proofs in classical arithmetic can also be given a dialectica interpretation via an initial embedding into Heyting arithmetic followed the dialectica interpretation of Heyting arithmetic.
• In constructive type theory, or in Heyting arithmetic extended with finite types, there is typically no separation at all-subsets of a type are given different treatments.
• From a constructivist point of view, Harrop formulae are " well-behaved . " For example, in Heyting arithmetic, Harrop formulae satisfy a classical equivalence not usually satisfied in constructive logic:
• Specifically, a statement is provable in Heyting arithmetic with Extended Church's thesis " and Markov's principle " if and only if there is a number that provably realizes it in Peano arithmetic.
• The formula translation describes how each formula A of Heyting arithmetic is mapped to a quantifier-free formula A _ D ( x; y ) of the system T, where x and y are tuples of fresh variables ( not appearing free in A ).
• The formula interpretation is such that whenever A is provable in Heyting arithmetic then there exists a sequence of closed terms t such that A _ D ( t; y ) is provable in the system T . The sequence of terms t and the proof of A _ D ( t; y ) are constructed from the given proof of A in Heyting arithmetic.
• The formula interpretation is such that whenever A is provable in Heyting arithmetic then there exists a sequence of closed terms t such that A _ D ( t; y ) is provable in the system T . The sequence of terms t and the proof of A _ D ( t; y ) are constructed from the given proof of A in Heyting arithmetic.
• If constructive arithmetic is translated using realizability into a classical meta-theory that proves the \ omega-consistency of the relevant classical theory ( for example, Peano Arithmetic if we are studying Heyting Arithmetic ), then Markov's principle is justified : a realizer is the constant function that takes a realization that P is not everywhere false to the unbounded search that successively checks if P ( 0 ), P ( 1 ), P ( 2 ), \ dots is true.