Existence Theorem

In mathematics, an existence theorem is a theorem with a statement beginning 'there exist(s) ..', or more generally 'for all x, y, ... there exist(s) ...'. That is, in more formal terms of symbolic logic, it is a theorem with a statement involving the existential quantifier. Many such theorems will not do so explicitly, as usually stated in standard mathematical language. For example, the statement that the sine function is continuous; or any theorem written in big O notation. The quantification can be found in the definitions of the concepts used.

A controversy that goes back to the early twentieth century concerns the issue of pure existence theorems. Such theorems may depend on non-constructive foundational material such as the axiom of infinity, the axiom of choice, or the law of excluded middle. From a constructivist viewpoint, by admitting them mathematics loses its concrete applicability (see nonconstructive proof). The opposing viewpoint is that abstract methods are far-reaching, in a way that numerical analysis cannot be.