Relative constructible universe can violate the continuum hypothesis

Work inside a transitive model $M$ of ZFC with $2^{{\aleph }_0}={\aleph }_2$. Choose a bijection $e:{\omega }_2^M\to \mathcal{P}(\omega {)}^M$ and encode its membership relation by a set of ordinals $A$. Use a canonical ordinal pairing that is definable in the constructible universe, rather than an arbitrary pairing available only in $M$: put the code of $(\xi ,n)$ into $A$ exactly when $n\in e(\xi )$.

The relative constructible universe $L(A)$ can decode every section $e(\xi )$, and hence contains every real number of $M$. Conversely $L(A)\subseteq M$, so the two models have the same sets of natural numbers. In particular they have the same ${\omega }_1$, because a countable ordinal is witnessed by a real coding a well-order. A bijection in $L(A)$ between its reals and ${\omega }_1$ would then also be such a bijection in $M$, contradicting the assumed cardinality of the real numbers. Thus $L(A)$ need not satisfy the Continuum hypothesis. The essential point is that a freely chosen parameter can carry an entire enumeration of the ground-model reals.