Difference between revisions of "Constructive proof"
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| − | A '''constructive proof''' demonstrates the existence of a [[mathematical]] function, number or object by producing (constructing) it. This is in contrast with other styles of proof, such as proof by contradiction, which asserts the existence of an object by finding a contradiction if it did not exist. Such a proof is called '''nonconstructive'''. | + | A '''constructive proof''' demonstrates the existence of a [[mathematical]] function, number or object by producing (constructing) it. This is in contrast with other styles of proof, such as proof by contradiction, which asserts the existence of an object by finding a contradiction if it did not exist. Such a proof is called '''nonconstructive''' and is not valued as highly by mathematicians, especially in [[applied mathematics]] and [[computer science]]. |
The [[Axiom of Choice]] assumes the existence of a function without constructing it, and thus all proofs that rely on the [[Axiom of Choice]] are nonconstructive proofs. | The [[Axiom of Choice]] assumes the existence of a function without constructing it, and thus all proofs that rely on the [[Axiom of Choice]] are nonconstructive proofs. | ||
Revision as of 00:26, August 7, 2008
A constructive proof demonstrates the existence of a mathematical function, number or object by producing (constructing) it. This is in contrast with other styles of proof, such as proof by contradiction, which asserts the existence of an object by finding a contradiction if it did not exist. Such a proof is called nonconstructive and is not valued as highly by mathematicians, especially in applied mathematics and computer science.
The Axiom of Choice assumes the existence of a function without constructing it, and thus all proofs that rely on the Axiom of Choice are nonconstructive proofs.
The easiest way to prove the existence of transcendental numbers is by a nonconstructive proof, arguing that the set of real numbers is uncountable while the set of algebraic numbers is countable, and thus (many) transcendental numbers must exist. Of course, finding a specific example is a much more difficult endeavor.