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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 non-constructive proof (also known as an existence proof or pure existence theorem ), which proves the existence of a particular kind of object ...
In classical real analysis, one way to define a real number is as an equivalence class of Cauchy sequences of rational numbers.. In constructive mathematics, one way to construct a real number is as a function ƒ that takes a positive integer and outputs a rational ƒ(n), together with a function g that takes a positive integer n and outputs a positive integer g(n) such that
From the other direction, there has been considerable clarification of what constructive mathematics is—without the emergence of a 'master theory'. For example, according to Errett Bishop's definitions, the continuity of a function such as sin(x) should be proved as a constructive bound on the modulus of continuity, meaning that the existential content of the assertion of continuity is a ...
Intuitionistic logic is a commonly-used tool in developing approaches to constructivism in mathematics. The use of constructivist logics in general has been a controversial topic among mathematicians and philosophers (see, for example, the Brouwer–Hilbert controversy). A common objection to their use is the above-cited lack of two central ...
The proof below is therefore given using the means of a constructive set theory. It is evident from the proof how the theorem relies on the axiom of pairing as well as an axiom of separation, of which there are notable variations. A crucial role in the set theoretic proof is also played by the axiom of extensionality. The subtleties the latter ...
The base logic of constructive analysis is intuitionistic logic, which means that the principle of excluded middle is not automatically assumed for every proposition.If a proposition . is provable, this exactly means that the non-existence claim . being provable would be absurd, and so the latter cannot also be provable in a consistent theory.
Pages which contain only proofs (of claims made in other articles) should be placed in the subcategory Category:Article proofs. Pages which contain theorems and their proofs should be placed in the subcategory Category:Articles containing proofs. Articles related to automatic theorem proving should be placed in Category:Automated theorem proving.
Constructive mathematics requires when proving "there exists an with property ()", one must construct a particular and a proof that it has property . In type theory, existence is accomplished using the dependent product type, and its proof requires a term of that type. An example of a non-constructive proof is proof by contradiction.