Mathematical logic and foundations Concept

Constructive proof

Builds on Existential quantifier Mathematical proof

Summary

A constructive proof of an existence statement $\exists x \, P(x)$ gives a specific object $a$ , or a method to find it, and proves $P(a)$ . A non-constructive proof shows that an object exists without giving it, for example by proof by contradiction.

Remarks

A classical non-constructive example: there are irrational numbers $a$ and $b$ such that $a^b$ is rational. Consider $\sqrt{2}^{\sqrt{2}}$ . It is rational or irrational, and each case gives such $a$ and $b$ . Intuitionistic logic accepts only constructive proofs of existence.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Proposition concept
Implication definition
Predicate definition
Modus ponens concept
Existential quantifier definition
Mathematical proof concept
Constructive proof concept

Where this leads

9 pages build on this one.

Tag	`01XM7JEKHY`
URI	`m2://01XM7JEKHY`
Kind	concept
Writing	W1 stated
MSC	03F65
Specializes	Mathematical proof
Part of	Proof methods
Uses	Existential quantifier, Mathematical proof
Same as	Wikidata Q3044470 Wikipedia Constructive proof

Constructive proof

Summary

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

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Proofs that use this method