Constructible universe ($L$)

The constructible universe is the inner model $L={\bigcup }_{\alpha \in \mathrm{Ord}}{L}_{\alpha }$ obtained by iterating the definable power set. It satisfies ZFC and the Generalized continuum hypothesis.

For $A\in L$, its constructible power set isUnlike the single-stage definable power set $\mathcal{D}(A)$, it includes subsets of $A$ appearing arbitrarily late in the constructible hierarchy.

The relative constructible universe $L(A)$ is obtained by running the constructible hierarchy over the transitive closure of a set $A$, retaining $A$ as a predicate or initial parameter. It is the smallest corresponding inner model containing the hereditary information coded by $A$.

Assume a transitive model $M$ satisfies $2^{{\aleph }_0}={\aleph }_2$. In $M$, encode a bijection $e:{\omega }_2\to \mathcal{P}(\omega )$ by one set $A\subseteq {\omega }_2$, using a fixed pairing of ${\omega }_2\times \omega$ with ${\omega }_2$. Then $L(A)$ decodes $e$ and contains every real of $M$. The two models consequently have the same ${\omega }_1$, and any bijection between ${\omega }_1$ and the real numbers in $L(A)$ would also be one in $M$. Thus $L(A)\models \neg \mathsf{CH}$.