Constructible power set (${\mathcal{P}}^L(A)$)

For $A\in L$, its constructible power set isUnlike the single-stage definable power set $\mathcal{D}(A)$, it includes subsets of $A$ appearing arbitrarily late in the constructible hierarchy.

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 ${\omega }_1^L\le {\omega }_1$ 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.

If the ambient power set of $\omega$ is not well-orderable and $a\in L$, then the full ambient power set of $a$ belongs to $L$ exactly when $a$ is finite. Finite set operations give the forward construction for finite $a$. If $a$ is infinite, its constructible well-order gives an injection $\omega \to a$. The induced injection of power sets would make the ambient power set of $\omega$ well-orderable if $\mathcal{P}(a)$ belonged to $L$.