Existence theorem

An existence theorem proves that at least one object or solution satisfies specified conditions, without necessarily giving a way to construct it.

Constructive proof: A proof that establishes a claim by explicitly producing an object or method. It proves existence by supplying the construction an existence theorem need not provide.
Constructive mathematics: A mathematical approach that accepts proofs according to constructive standards of evidence. It challenges existence arguments that establish no explicit witness.
Fundamental theorem of algebra: The theorem that every nonconstant complex polynomial has at least one complex root. It guarantees a root even when a particular polynomial's roots are not immediately known.
Existential quantifier: A logical operator asserting that at least one object in a domain satisfies a predicate. It is the logical form an existence theorem establishes.
Proof by contradiction: A proof that derives a contradiction from the assumption that a claim is false. It can establish existence indirectly by ruling out the possibility that no object qualifies.
Uniqueness theorem: A theorem proving that exactly one object satisfies specified conditions. It strengthens existence by ruling out all but one qualifying object.
Weierstrass extreme value theorem: The theorem that a continuous real-valued function on a compact set attains a maximum and minimum. It guarantees an optimizer under continuity and compactness conditions.
Predicate (mathematical logic): A statement with variables that becomes true or false when values are assigned to them. The theorem's conditions specify the predicate an existing object must satisfy.
Compactness theorem: A theorem connecting finite satisfiability with satisfiability for collections of logical statements. Compactness can turn finite consistency into the existence of a model.
Decision problem: A problem asking whether a given input has a specified yes-or-no property. Knowing that a solution exists does not necessarily decide how to find or recognize one.

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