= 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.
Back to article page
