The Constructible Hierarchy is the Constructible Model of Set Theory:
Also, by Consistency of Choice:
Furthermore is Transitive and Axiom of Constructability is Satisfied:
We now prove stuff inside of .
Lemma (main idea)
For every , , there is some such that .
Proof
We work in , which is a Model of .
Let and note that this is a Transitive set.
Find large enough by Lévy Reflection Theorem
such that and is absolute between and
where is from Gödel’s Condensation Lemma.
Because , we get .
Now using The Downward Löwenheim-Skolem Theorem
find countable such that with Absolute between and .
So and is countable.
Form the Transitive by Mostowski’s Collapsing Theorem.
Then by Gödel’s Condensation Lemma, there is some such that
Since is countable, so is .
But , so it has to be .
Theorem
The Continuum Hypothesis holds in :
Proof
By the previous lemma, all subsets of are contained in .
Furthermore, .
Thus the powerset of is of cardinality at most .