Levy's lemma on hereditary cardinality

Question

Let $\kappa$ be a cardinal and recall $HC(\kappa^+)= \{y: |TC(y)|<\kappa^+\}$.

If $x\in HC(\kappa^+)$ and $x\in L_{\alpha}$ for some $\alpha$, then $x\in L_{\beta}$ for some $\beta \in HC(\kappa^+)$, where $L$ denotes the constructible hierarchy.

I saw this result in a proof of the consistency of GCH but I don't remember seeing it proved beforehand. It was attributed to Levy for what it's worth.

Could someone please point me to a proof or provide me one directly? The more basic and more specific the proof, the better.

Answer 1

This is a direct application of the condensation lemma. Let me assume that without loss of generality, $\alpha$ is a limit ordinal.

Since $x\in L_\alpha$, consider the elementary submodel of $L_\alpha$ generated by $\operatorname{TC}(\{x\})$, this is some model $N$ such that:

1. $N$ is well-founded,
2. $x\in N$,
3. the transitive closure of $x$ is a subset of $N$, and
4. $|N|\leq|\operatorname{TC}(x)|+\aleph_0$.

Apply the Mostowski collapse lemma, and we get some transitive $N'$ such that $x\in N'$ (this is because of 3 in the above list).

But, we know that $N\models V=L$, so $N'\models V=L$. And the only transitive models of $V=L$ are of the form $L_\beta$. So $N'=L_\beta$ for some $\beta$.

Finally, since $|N|<\kappa^+$, it follows that $\beta<\kappa^+$ as well, since $|N'|=|\beta|$.