Non-constructive proofs[edit]

First consider the theorem that there are an infinitude of prime numbers. Euclid's proof is constructive. But a common way of simplifying Euclid's proof postulates that, contrary to the assertion in the theorem, there are only a finite number of them, in which case there is a largest one, denoted n. Then consider the number n! + 1 (1 + the product of the first n numbers). Either this number is prime, or all of its prime factors are greater than n. Without establishing a specific prime number, this proves that one exists that is greater than n, contrary to the original postulate.


Now consider the theorem "there exist irrational numbers ${\displaystyle a}$ and ${\displaystyle b}$ such that ${\displaystyle a^{b}}$ is rational." This theorem can be proven by using both a constructive proof, and a non-constructive proof.


The following 1953 proof by Dov Jarden has been widely used as an example of a non-constructive proof since at least 1970:^[4][5]