= Solution
Use the <constructible universe theorem> that $L$ satisfies $ZFC$ and <GCH>. In particular, internally it has a <bijection>
$$
b:\omega_1^L\longrightarrow\mathcal P(\omega)^L.
$$
This <bijection> belongs to $L$ and is also an actual <bijection> in the ambient universe, because its domain, range and graph are transitive-model sets with the same membership relation.
We must compare the <ordinals> rather than assume <cardinality> is absolute. Every <ordinal> below $\omega_1^L$ is <countable> in $L$, and its witnessing enumeration belongs to the ambient universe too. Thus every such <ordinal> is <countable> externally, proving $\omega_1^L\leq\omega_1$. Consequently the ambient <cardinality> of the constructible reals is at most $\aleph_1$.
If every subset of $\omega$ were constructible, the displayed <bijection> would enumerate the full ambient <power set> using at most $\aleph_1$ <points>. This contradicts the assumption $2^{\aleph_0}=\aleph_2$. Therefore
$$
\boxed{\exists X\subseteq\omega\quad X\notin L.}
$$
The argument permits $\omega_1^L$ to be strictly smaller than the ambient $\omega_1$, exactly the <cardinal>-absoluteness issue highlighted in the source.
Back to article page
