Existence theorem

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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.

'Pure' existence results[edit]

An existence theorem may be called pure if the proof given of it doesn't also indicate a construction of whatever kind of object the existence of which is asserted.

From a more rigorous point of view, this is a problematic concept. This is because it is a tag applied to a theorem, but qualifying its proof; hence, pure is here defined in a way which violates the standard proof irrelevance of mathematical theorems. That is, theorems are statements for which the fact is that a proof exists, without any 'label' depending on the proof: they may be applied without knowledge of the proof, and indeed if that's not the case the statement is faulty. Thus, many constructivist mathematicians work in extended logics (such as intuitionistic logic) where pure existence statements are intrinsically weaker than their constructivist counterparts.

Such pure existence results are in any case ubiquitous in contemporary mathematics. For example, for a linear problem the set of solutions will be a vector space, and some a priori calculation of its dimension may be possible. In any case where the dimension is probably at least 1, an existence assertion has been made (that a non-zero solution exists.)

Theoretically, a proof could also proceed by way of a metatheorem, stating that a proof of the original theorem exists (for example, that a proof by exhaustion search for a proof would always succeed). Such theorems are relatively unproblematic when all of the proofs involved are constructive; however, the status of "pure existence metatheorems" is extremely unclear.

Constructivist ideas[edit]

From the other direction there has been considerable clarification of what constructive mathematics is; without the emergence of a 'master theory'. For example according to Errett Bishop's definitions, the continuity of a function (such as sin x) should be proved as a constructive bound on the modulus of continuity, meaning that the existential content of the assertion of continuity is a promise that can always be kept. One could get another explanation from type theory, in which a proof of an existential statement can come only from a term (which we can see as the computational content).