# Category: Constructivism (mathematics)

Constructive analysis
In mathematics, constructive analysis is mathematical analysis done according to some principles of constructive mathematics.This contrasts with classical analysis, which (in this context) simply mean
Computable analysis
In mathematics and computer science, computable analysis is the study of mathematical analysis from the perspective of computability theory. It is concerned with the parts of real analysis and functio
Brouwer–Hilbert controversy
In a controversy over the foundations of mathematics, in twentieth-century mathematics, L. E. J. Brouwer, a proponent of the constructivist school of intuitionism, opposed David Hilbert, a proponent o
Intuitionistic type theory
Intuitionistic type theory (also known as constructive type theory, or Martin-Löf type theory) is a type theory and an alternative foundation of mathematics.Intuitionistic type theory was created by P
Primitive recursive arithmetic
Primitive recursive arithmetic (PRA) is a quantifier-free formalization of the natural numbers. It was first proposed by Norwegian mathematician , as a formalization of his finitist conception of the
Constructive set theory
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
Heyting arithmetic
In 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.
Disjunction and existence properties
In mathematical logic, the disjunction and existence properties are the "hallmarks" of constructive theories such as Heyting arithmetic and constructive set theories (Rathjen 2005).
Computable model theory
Computable model theory is a branch of model theory which deals with questions of computability as they apply to model-theoretical structures. Computable model theory introduces the ideas of computabl
Modulus of convergence
In real analysis, a branch of mathematics, a modulus of convergence is a function that tells how quickly a convergent sequence converges. These moduli are often employed in the study of computable ana
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 classicall
Construction of the real numbers
In mathematics, there are several equivalent ways of defining the real numbers. One of them is that they form a complete ordered field that does not contain any smaller complete ordered field. Such a
Indecomposability (intuitionistic logic)
In intuitionistic analysis and in computable analysis, indecomposability or indivisibility (German: Unzerlegbarkeit, from the adjective unzerlegbar) is the principle that the continuum cannot be parti
Kőnig's lemma
Kőnig's lemma or Kőnig's infinity lemma is a theorem in graph theory due to the Hungarian mathematician Dénes Kőnig who published it in 1927. It gives a sufficient condition for an infinite graph to h
Choice sequence
In intuitionistic mathematics, a choice sequence is a constructive formulation of a sequence. Since the Intuitionistic school of mathematics, as formulated by L. E. J. Brouwer, rejects the idea of a c
Bar induction
Bar 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 us
Finitist set theory
Finitist set theory (FST) is a collection theory designed for modeling finite nested structures of individuals and a variety of transitive and antitransitive chains of relations between individuals. U
Non-constructive algorithm existence proofs
The vast majority of positive results about computational problems are constructive proofs, i.e., a computational problem is proved to be solvable by showing an algorithm that solves it; a computation
Brouwer–Heyting–Kolmogorov interpretation
In mathematical logic, the Brouwer–Heyting–Kolmogorov interpretation, or BHK interpretation, of intuitionistic logic was proposed by L. E. J. Brouwer and Arend Heyting, and independently by Andrey Kol
Minimal logic
Minimal logic, or minimal calculus, is a symbolic logic system originally developed by Ingebrigt Johansson. It is an intuitionistic and paraconsistent logic, that rejects both the law of the excluded
Limited principle of omniscience
In 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 t
Realizability
In mathematical logic, realizability is a collection of methods in proof theory used to study constructive proofs and extract additional information from them. Formulas from a formal theory are "reali
Church's thesis (constructive mathematics)
In constructive mathematics, Church's thesis is an axiom stating that all total functions are computable functions. thus restricts the class of functions to computable ones and consequently is incompa
Friedman translation
In mathematical logic, the Friedman translation is a certain transformation of intuitionistic formulas. Among other things it can be used to show that the Π02-theorems of various first-order theories
Pseudo-order
In constructive mathematics, a pseudo-order is a constructive generalisation of a linear order to the continuous case. The usual trichotomy law does not hold in the constructive continuum because of i
Harrop formula
In intuitionistic logic, the Harrop formulae, named after , are the class of formulae inductively defined as follows: * Atomic formulae are Harrop, including falsity (⊥); * is Harrop provided and ar
Constructive proof
In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a n
Axiom schema of predicative separation
In axiomatic set theory, the axiom schema of predicative separation, or of restricted, or Δ0 separation, is a schema of axioms which is a restriction of the usual axiom schema of separation in Zermelo
Constructivism (philosophy of mathematics)
In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrasting
Ultrafinitism
In the philosophy of mathematics, ultrafinitism (also known as ultraintuitionism, strict formalism, strict finitism, actualism, predicativism, and strong finitism) is a form of finitism and intuitioni
Apartness relation
In constructive mathematics, an apartness relation is a constructive form of inequality, and is often taken to be more basic than equality. It is often written as (⧣ in unicode) to distinguish from th
Heyting field
A Heyting field is one of the inequivalent ways in constructive mathematics to capture the classical notion of a field. It is essentially a field with an apartness relation. A commutative ring is a He
Inhabited set
In constructive mathematics, a set is inhabited if there exists an element In classical mathematics, this is the same as the set being nonempty; however, this equivalence is not valid in intuitionisti
Diaconescu's theorem
In mathematical logic, Diaconescu's theorem, or the Goodman–Myhill theorem, states that the full axiom of choice is sufficient to derive the law of the excluded middle, or restricted forms of it, in c
Intuitionistic logic
Intuitionistic 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
Subcountability
In 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 fro
Constructive nonstandard analysis
In mathematics, constructive nonstandard analysis is a version of Abraham Robinson's nonstandard analysis, developed by Moerdijk (1995), Palmgren (1998), Ruokolainen (2004). Ruokolainen wrote: The pos
Intuitionism
In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activ
Modulus of continuity
In mathematical analysis, a modulus of continuity is a function ω : [0, ∞] → [0, ∞] used to measure quantitatively the uniform continuity of functions. So, a function f : I → R admits ω as a modulus o
Heyting algebra
In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped w
Finitism
Finitism is a philosophy of mathematics that accepts the existence only of finite mathematical objects. It is best understood in comparison to the mainstream philosophy of mathematics where infinite m