Let $T$ be the set of all transcendental numbers. Show |$T$| = |$\Bbb R$|.
Now $T$ is a subset of $\Bbb R$ so clearly |$T$| $\le$|$\Bbb R$|.
But we can also show that $T$ is uncountable. Suppose by contradiction that $T$ is countable. Let $A$ denote the set of all algebraic numbers. I have shown elsewhere that $A$ is countable. Now $A\cup T$ = R. Since the union of two countable sets is countable, $A\cup T$ is countable, contradicting that R is uncountable. Hence $T$ is uncountable.
Okay so now I'm basically stuck in the undecidability of the continuum hypothesis because I have|$\Bbb N$| < |$T$| $\le$ |$\Bbb R$| = $c$.
Help?
