Hi I hope this question is accurate. Gödel and Cohen could show that the Continuum Hypothesis (CH) is independent from ZFC using models in which CH holds, and fails respectively. My question now is:

If we take a large enough part of the universe, say $V_{\omega_{\omega}}$ , does CH hold in it? And why can't we go on to argue like this:

If $V_{\omega_{\omega}} \models CH$ then there is a bijection between $\omega_1$ and $2^{\aleph_0}$, and as $V_{\omega_{\omega}}$ is transitive this is 'really' a bijection thus CH holds in ZFC.

If $V_{\omega_{\omega}} \models \lnot CH$ then, as any possible bijection between $\omega_1$ and $2^{\omega}$ is already an element of $V_{\omega_{\omega}}$, CH is inside ZFC refuteable?

This are maybe crude argumentations but I can't answer that by myself.