nonconstructive

Nonconstructive refers to proofs or arguments in mathematics that establish the existence of a mathematical object or truth without providing an explicit example or a method to construct the object. Nonconstructive reasoning often relies on meta-mathematical principles such as the law of excluded middle, proof by contradiction, or the use of nonconstructive existence theorems.

In contrast, constructive proofs yield an explicit example or a procedure to construct the object, often with an algorithm or finite steps. In constructive or computable mathematics, existence claims require a construction.

Common sources of nonconstructive proofs include combinatorial arguments that only show existence via counting or extremal principles, proofs using the compactness theorem, or arguments that invoke the axiom of choice (or Zorn's lemma) to assert the existence of an object without giving a method to obtain it.

Classic examples include: the existence of irrational numbers a and b such that a^b is rational, proven by considering x = sqrt(2)^sqrt(2) and analyzing cases; if sqrt(2)^sqrt(2) is rational, done; if irrational, then (sqrt(2)^sqrt(2))^sqrt(2) = 2 is rational. Another broad example is the existence of a basis for every vector space (over any field) or a nonprincipal ultrafilter, guaranteed by the axiom of choice, without explicit construction.

Nonconstructive results are central in some areas of analysis, model theory, and set theory, and they illustrate a divergence between existence and explicit computability. Critics argue they offer less practical value, while supporters note they reveal what is mathematically inevitable even when explicit methods are unavailable.