Definition:Nonconstructive Proof

Definition

A nonconstructive proof is a proof in which there does not exist an effective procedure for the construction of every object in it.

That is, such that it requires an infinite number of steps to complete.

Examples

Natural Number

This is the shape of a proof which is nonconstructive:

It is false that every number $n$ lacks the property $P$.

Therefore there exists at least one number $n_0$ that has property $P$.

Unless an example of such a $n_0$ can be constructed, such an argument is not allowed in a constructive proof.

Also known as

Nonconstructive proof can also be used hyphenated: non-constructive proof, but house style prefers the non-hyphenated version.

Also see

• Axiom:Axiom of Choice
• Definition:Constructive Proof
• Definition:Intuitionism

• Results about nonconstructive proofs can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): intuitionism
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): intuitionism
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): nonconstructive proof