 Solution

Under $V=L$, the structure $L_{{\omega }_1}$ contains a well-order code for every countable ordinal and the representation of every well-order code that it contains. Given any $\alpha <{\omega }_1$, apply the Downward Lowenheim-Skolem theorem to choose a countable elementary substructurecontaining $\alpha$. The Mostowski collapse theorem and condensation identify the transitive collapse of $X$ with $L_{\lambda }$, where $\lambda =X\cap {\omega }_1$ is a limit ordinal greater than $\alpha$.

Elementarity now verifies both coding properties. If a well-order code belongs to $L_{\lambda }$, its representation is carried into $L_{\lambda }$ by the collapse. Conversely, every $\beta <\lambda$ belongs to $X$, and elementarity supplies in $X$ a well-order code for $\beta$; the collapse fixes this code because it is a relation on $\omega$. Hence $L_{\lambda }$ is a coding level of the constructible hierarchy. Such $\lambda$ occur unboundedly below ${\omega }_1$, so $\Gamma$ has at least ${\aleph }_1$ elements; since $\Gamma \subseteq {\omega }_1$, it has exactly

Solved by gpt-5.6-sol high.