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This is in contrast to a non-constructive proof (also known as an existence proof or pure existence theorem), which proves the existence of a particular kind of object without providing an example. For avoiding confusion with the stronger concept that follows, such a constructive proof is sometimes called an effective proof .
Constructive versus non-constructive [ edit ] As is often the case with probabilistic arguments, this theorem is nonconstructive and gives no method of determining an explicit element of the probability space in which no event occurs.
For example, any theorem of classical propositional logic of the form has a proof consisting of an intuitionistic proof of followed by one application of double-negation elimination. Intuitionistic logic can thus be seen as a means of extending classical logic with constructive semantics.
An interactive proof session in CoqIDE, showing the proof script on the left and the proof state on the right. Coq is an interactive theorem prover first released in 1989. It allows for expressing mathematical assertions, mechanically checks proofs of these assertions, helps find formal proofs, and extracts a certified program from the constructive proof of its formal specification.
The example mapping f happens to correspond to the example enumeration s in the picture above. A generalized form of the diagonal argument was used by Cantor to prove Cantor's theorem: for every set S, the power set of S—that is, the set of all subsets of S (here written as P(S))—cannot be in bijection with S itself. This proof proceeds as ...
A waiter in California has been fired after reportedly asking a group of customers for 'proof of residency.'
The proof given above is not generally considered to be constructive, because at each step it uses a proof by contradiction to establish that there exists an adjacent vertex from which infinitely many other vertices can be reached, and because of the reliance on a weak form of the axiom of choice.
constructive proof A proof that demonstrates the existence of a mathematical object by providing a method to construct it explicitly, as opposed to proving indirectly by contradiction. contextualism The theory that the context in which an assertion is made is essential for understanding its truth conditions and meaning. contingency