Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 121 / 2 / iv

Take a transitive model $M\models \mathsf{ZFC}+2^{{\aleph }_0}={\aleph }_2$. In $M$, choose a bijection $e:{\omega }_2\to \mathcal{P}(\omega )$ and encode its graph by a set $A\subseteq {\omega }_2$, using a fixed bijection between ${\omega }_2\times \omega$ and ${\omega }_2$. The relative constructible universe $L(A)$ can decode $e$, and therefore contains every real number of $M$; being an inner model of $M$, it has no additional reals.

Models with the same reals have the same first uncountable ordinal, because their reals code exactly the same countable well-orders. If $L(A)$ satisfied the Continuum hypothesis, its bijection between ${\omega }_1$ and the reals would also belong to $M$, contradicting $M\models 2^{{\aleph }_0}={\aleph }_2$. This is the construction in relative constructible universe can violate the continuum hypothesis, and it gives