Μ operatorIn computability theory, the μ-operator, minimization operator, or unbounded search operator searches for the least natural number with a given property. Adding the μ-operator to the primitive recursive functions makes it possible to define all computable functions. Suppose that R(y, x1, ..., xk) is a fixed (k+1)-ary relation on the natural numbers. The μ-operator "μy", in either the unbounded or bounded form, is a "number theoretic function" defined from the natural numbers to the natural numbers.
UltrafinitismIn the philosophy of mathematics, ultrafinitism (also known as ultraintuitionism, strict formalism, strict finitism, actualism, predicativism, and strong finitism) is a form of finitism and intuitionism. There are various philosophies of mathematics that are called ultrafinitism. A major identifying property common among most of these philosophies is their objections to totality of number theoretic functions like exponentiation over natural numbers.
Limited principle of omniscienceIn constructive mathematics, the limited principle of omniscience (LPO) and the lesser limited principle of omniscience (LLPO) are axioms that are nonconstructive but are weaker than the full law of the excluded middle. They are used to gauge the amount of nonconstructivity required for an argument, as in constructive reverse mathematics. These principles are also related to weak counterexamples in the sense of Brouwer. The limited principle of omniscience states : LPO: For any sequence , , ...
Bar inductionBar induction is a reasoning principle used in intuitionistic mathematics, introduced by L. E. J. Brouwer. Bar induction's main use is the intuitionistic derivation of the fan theorem, a key result used in the derivation of the uniform continuity theorem. It is also useful in giving constructive alternatives to other classical results. The goal of the principle is to prove properties for all infinite sequences of natural numbers (called choice sequences in intuitionistic terminology), by inductively reducing them to properties of finite lists.
SubcountabilityIn constructive mathematics, a collection is subcountable if there exists a partial surjection from the natural numbers onto it. This may be expressed as where denotes that is a surjective function from a onto . The surjection is a member of and here the subclass of is required to be a set. In other words, all elements of a subcountable collection are functionally in the image of an indexing set of counting numbers and thus the set can be understood as being dominated by the countable set .
Calculus of constructionsIn mathematical logic and computer science, the calculus of constructions (CoC) is a type theory created by Thierry Coquand. It can serve as both a typed programming language and as constructive foundation for mathematics. For this second reason, the CoC and its variants have been the basis for Coq and other proof assistants. Some of its variants include the calculus of inductive constructions (which adds inductive types), the calculus of (co)inductive constructions (which adds coinduction), and the predicative calculus of inductive constructions (which removes some impredicativity).
