Intuitionism

Remarks

The statement "there exist irrational numbers $a$ and $b$ with $a^b$ rational" has an intuitionistic proof. Take $a = \sqrt 2$ and $b = 2 \log_2 3$; then $a^b = 2^{\log_2 3} = 3$. Both numbers are shown to be irrational directly. The classical proof considers the two cases "$\sqrt 2^{\sqrt 2}$ is rational" and "it is irrational". This proof is not intuitionistic: it uses excluded middle and does not say which case holds. The logic of intuitionism was formalized by Arend Heyting as intuitionistic logic, and its meaning of the connectives is the Brouwer–Heyting–Kolmogorov interpretation.