Effective topos

In mathematics, the effective topos introduced by captures the mathematical idea of effectivity within the framework. The topos is based on the partial combinatory algebra given by Kleene's first algebra . In Kleene's notion of recursive realizability, any predicate is assigned realizing numbers, i.e. a subset of . The extremal propositions are and , realized by and . However in general, this process assigns more data to a proposition than just a binary truth value. A formula with free variables will give rise to a map in the values of which is the subset of corresponding realizers. is a prime example of a realizability topos. These are a class of elementary topoi with an intuitionistic internal logic and fulfilling a form of dependent choice. They are generally not Grothendieck topoi. In particular, the effective topos is . Other realizability topos construction can be said to abstract away the some aspects played by here. The objects are pairs of sets together with a symmetric and transitive relation in , representing a form of equality. But the so called existence predicate can be empty, and so this is not generally reflexive. Arrows amount to equivalence classes of functional relations appropriately respecting the defined equalities. The classifier amounts to . The pair (or, by abuse of notation, just that underlying powerset) may be denoted as . An entailment relation on makes it into a Heyting pre-algebra. Such a context allows to define the appropriate lattice-like logic structure, with logical operations given in terms of operations of the realizer sets, making use of pairs and computable functions. The terminal object is a singleton with trivial existence predicate (). The finite product respect the equality appropriately. The classifier's equality is given through equivalences in its lattice. Some objects exhibit a rather trivial existence predicate depending only on the validity of the equality relation "" of sets, so that valid equality maps to the top set and rejected equality maps to .

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Related concepts (5)

Markov's principle

Markov'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.

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.

Constructive set theory

Axiomatic 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.