Do we need Generalized Continuum Hypothesis to prove that $2^{2^{\aleph_0}}>2^{\aleph_0}$

Question

Do we need Generalized Continuum Hypothesis to prove that

$$2^{2^{\aleph_0}}>2^{\aleph_0}$$

In other words, the cardinality of $ 2^{2^{\aleph_0}}$ is strictly greater than the cardinality of $2^{\aleph_0}$ ? Do we need Generalized Continuum Hypothesis to do it?

Answer 1

No, you don't. Cantor's theorem is the statement that $2^\kappa > \kappa$ for every cardinal $\kappa$. It is a theorem of ZF.

(In ZFC it is clear what $>$ means, but in ZF there can be a little question. Cantor's theorem proves there is no surjection from $\kappa$ to $2^\kappa$. But it's also true in ZF that there is no injection from $2^\kappa$ to $\kappa$. If there were, then since there is an obvious injection from $\kappa$ to $2^{\kappa}$, the Cantor-Schroeder-Bernstein theorem, also valid in ZF, would show there is a bijection from $\kappa$ to $2^{\kappa}$, which in particular would be a surjection and contradict Cantor's theorem.)

Answer 2

Cantor's theorem (with the standard diagonalisation proof) states in general that $2^\kappa > \kappa$ for any cardinal $\kappa$. So there is no need for assumptions like CH or GCH to conclude this, only ZF.

The generalized continuum hypothesis tells you how much bigger $2^{2^{\aleph_0}}$ is than $2^{\aleph_0}$ (it's the next cardinality), but it's not necessary to prove that it's strictly larger.