---
tag: 01XM7JEKHY
kind: concept
name: Constructive proof
msc: [03F65]
uses: [0JW2FHW7G3, 0B94TKFQJX]
sources:
  - url: https://en.wikipedia.org/wiki/Constructive_proof
    locator: lead section
same_as:
  wikidata: Q3044470
  wikipedia: Constructive 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|0J0XBKG64S]].

## 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.

## Explanations

### L1 human

A constructive proof that a solution exists shows you the solution.
A non-constructive proof only shows that "no solution" is impossible.
Both are correct in classical mathematics, but the constructive proof tells you more.