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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 ...
Informally, this means that if there is a constructive proof that an object exists, that constructive proof may be used as an algorithm for generating an example of that object, a principle known as the Curry–Howard correspondence between proofs and algorithms.
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
Element (mathematics) Ur-element; Singleton (mathematics) Simple theorems in the algebra of sets; Algebra of sets; Power set; Empty set; Non-empty set; Empty function; Universe (mathematics) Axiomatization; Axiomatic system. Axiom schema; Axiomatic method; Formal system; Mathematical proof. Direct proof; Reductio ad absurdum; Proof by ...
In mathematical logic, realizability is a collection of methods in proof theory used to study constructive proofs and extract additional information from them. [1] Formulas from a formal theory are "realized" by objects, known as "realizers", in a way that knowledge of the realizer gives knowledge about the truth of the formula. There are many ...
A philosophy of mathematics that denies the reality of the mathematical infinite and the completeness of mathematical truth, requiring constructive proofs. intuitionistic logic A system of logic that reflects the principles of intuitionism, rejecting the law of excluded middle and requiring more constructive proofs of existence.