= Relative constructible universe can violate the continuum hypothesis
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 $L(A)\models\neg\mathsf{CH}$.
Back to article page
