TITLE: About the proof of the proposition "there exists irrational numbers a, b such that a^b is rational"
QUESTION [6 upvotes]: What does the classical proof of the proposition "there exists irrational numbers a, b such that $a^b$ is rational" want to reveal? I know it has something to do with the difference between classical and constructive mathematics but am not quite clear about it. Materials I found online does not give quite clear explanations either. Could someone give a better explanation?

REPLY [12 votes]: The proposition can also be proved without explicitly knowing any particular irrational number:
Fix a positive rational number $c$. Then for any positive irrational real number number $a$,
the equation $a^b = c$ has a solution, and the solutions for different $a$ are different.
Now assume that there would be no irrational numbers $a$ and $b$ such that $a^b$ is rational.
Then all these solutions would be rational, thus we would have an injective mapping from the
positive irrational real numbers to the rationals; however we know that the rationals are
countable and the positive irrational reals are not, so there is no such injection, and we
have a contradiction.