1.1 Non-constructive and constructive proofs
Mathematicians commonly use an indirect method of proof called non-constructive proof: they prove the existence of an object without presenting (constructing) the object. However, many times they can also use a direct method of proof called constructive proof: to prove the existence of an object by presenting (constructing) the object.
A non-constructive proof is a proof that proves the existence of an object without presenting the object.
A constructive proof is a proof that proves the existence of an object by presenting the object.
From a logical point of view, a non-constructive proof uses the law of excluded middle while a constructive proof does not use the law of excluded middle.
Definition 2. The law of excluded middle is the assertion “every statement is true or false”.
To illustrate this discussion, let us see the usual example of a theorem with non-constructive and constructive proofs.
Theorem 3. There are irra...