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The above proof is an example of a non-constructive proof disallowed by intuitionists: The proof is non-constructive because it doesn't give specific numbers a {\\displaystyle a} and b {\\displaystyle b} that satisfy the theorem but only two separate possibilities, one of which must work.
However, there is a non-constructive proof that shows that linkedness is decidable in polynomial time. The proof relies on the following facts: The set of graphs for which the answer is "yes" is closed under taking minors. I. e., if a graph G can be embedded linklessly in 3-d space, then every minor of G can also be embedded linklessly.
At its core, this proof is non-constructive because it relies on the statement "Either q is rational or it is irrational"—an instance of the law of excluded middle, which is not valid within a constructive proof. The non-constructive proof does not construct an example a and b; it merely gives a number of possibilities (in this case, two ...
A non-constructive proof might show a solution exists without specifying either an algorithm to obtain it or a specific bound. Even if the proof is constructive, showing an explicit bounding polynomial and algorithmic details, if the polynomial is not very low-order the algorithm might not be sufficiently efficient in practice.
An example of a non-constructive proof is proof by contradiction. The first step is assuming that does not exist and refuting it by contradiction. The conclusion from that step is "it is not the case that does not exist". The last step is, by double negation, concluding that exists.
In contrast, a constructive proof establishes that a particular object exists by providing a method of finding it. The following famous example of a nonconstructive proof shows that there exist two irrational numbers a and b such that a b {\displaystyle a^{b}} is a rational number .
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.
Another aspect of this article had a great impact on mathematics of the 20th century; this is the systematic use of non-constructive methods. For example, the basis theorem asserts that every ideal has a finite generator set, but the original proof does not provide any way to compute it for a specific ideal.