The best way to do that is to prove that its complement in the set of real numbers, the set of "algebraic numbers" is countable.

A number is algebraic (of order n) if and only if it satisfies a polynomial equation with integer coefficients, of degree n, and no such polynomial of lower degree.

Of course, for any positive integer n, there exist only a countable number of polynomials of degree n and each has, at most, n solutions. Thus, there exist a countable number of numbers that are algebraic of degree n. Further, the set of all algebraic numbers is the union of "numbers that are algebraic of degree n" for all n.