= Ambient cardinal bound on the constructible reals
{title2=$|\mathcal P(\omega)\cap L|\leq\aleph_1$}
In a <ZFC> universe, the <constructible universe theorem> supplies a <bijection> inside $L$ between $\omega_1^L$ and its reals. The same graph is a <bijection> externally. Each <ordinal> below $\omega_1^L$ has a countability witness in $L$, hence also externally, proving $\omega_1^L\leq\omega_1$ without assuming <cardinal> absoluteness. Therefore the constructible reals have ambient <cardinality> at most $\aleph_1$. If the ambient continuum is larger, some subset of $\omega$ is not constructible. The <ordinal> $\omega_1^L$ may be externally <countable>, so the argument is an upper bound and not a claim of equality.
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