Church's thesis (constructive mathematics)In constructive mathematics, Church's thesis is an axiom stating that all total functions are computable functions. The similarly named Church–Turing thesis states that every effectively calculable function is a computable function, thus collapsing the former notion into the latter. is stronger in the sense that with it every function is computable. The constructivist principle is fully formalizable, using formalizations of "function" and "computable" that depend on the theory considered.
Markov's principleMarkov's principle, named after Andrey Markov Jr, is a conditional existence statement for which there are many equivalent formulations, as discussed below. The principle is logically valid classically, but not in intuitionistic constructive mathematics. However, many particular instances of it are nevertheless provable in a constructive context as well. The principle was first studied and adopted by the Russian school of constructivism, together with choice principles and often with a realizability perspective on the notion of mathematical function.
Dialectica interpretationIn proof theory, the Dialectica interpretation is a proof interpretation of intuitionistic logic (Heyting arithmetic) into a finite type extension of primitive recursive arithmetic, the so-called System T. It was developed by Kurt Gödel to provide a consistency proof of arithmetic. The name of the interpretation comes from the journal Dialectica, where Gödel's paper was published in a 1958 special issue dedicated to Paul Bernays on his 70th birthday.
Constructive set theoryAxiomatic constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory. The same first-order language with "" and "" of classical set theory is usually used, so this is not to be confused with a constructive types approach. On the other hand, some constructive theories are indeed motivated by their interpretability in type theories. In addition to rejecting the principle of excluded middle (), constructive set theories often require some logical quantifiers in their axioms to be set bounded, motivated by results tied to impredicativity.
Heyting arithmeticIn 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 can be characterized just like the first-order theory of Peano arithmetic , except that it uses the intuitionistic predicate calculus for inference. In particular, this means that the double-negation elimination principle, as well as the principle of the excluded middle , do not hold.
Intuitionistic logicIntuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not assume the law of the excluded middle and double negation elimination, which are fundamental inference rules in classical logic. Formalized intuitionistic logic was originally developed by Arend Heyting to provide a formal basis for L.
