Gödel's consistency of CH and AC

Remarks

The construction of $L$ can be carried out inside any model of ZF, and it is absolute, so $L$ computed inside $L$ is $L$ again; this gives part 1. The axiom of choice holds in $L$ because $L$ has a definable well-order. A set enters at a least stage and is defined there by a formula with parameters, and these data can be ordered. For the continuum hypothesis, assume $V = L$; the condensation lemma shows that every subset of $\omega_\alpha$ lies in $L_{\omega_{\alpha + 1}}$, a set of size $\aleph_{\alpha + 1}$. The result is a relative consistency proof: by Gödel's second incompleteness theorem, ZF cannot prove its own consistency, so the hypothesis "ZF is consistent" cannot be removed. Together with Cohen's theorem, it shows that the continuum hypothesis is independent of ZFC.

Going deeper

For a student

The theorem is the first half of the independence of the continuum hypothesis; the second half is Cohen's forcing argument. The method is the method of inner models: one finds a definable class in which the axioms hold together with the desired statement. Inner models alone cannot show that the continuum hypothesis can fail: if $V = L$, then $L$ is the only inner model, and the continuum hypothesis holds in it. To show that the continuum hypothesis can fail, one needs a way to enlarge the universe, and this is what forcing does.