I'm working through Kunen (1980 version) and there is an exercise (pg. 180) to prove that under $V=L$, $L(\kappa) = H(\kappa)$ for all infinite cardinals $\kappa$ (where $H(\kappa)=\{x:|\operatorname{trcl}(x)|<\kappa\}).$
I wrote a proof but I suspect it is too simple and something is wrong. The reason I think it is too simple is that the text proves the theorem for the special case where $\kappa$ is regular. This proof seems marginally easier than that one and doesn't use it. Also, it is proven in Kunen's newer edition (2013-Thm II.6.23) and while the proof is similar to mine below, it looks significantly more complicated. Here is the proof I gave:
Proof. Kunen's proof of $L(\kappa)\subseteq H(\kappa)$ doesn't depend on the regularity of $\kappa$ (or V=L for that matter) so what is left is $H(\kappa)\subseteq L(\kappa).$ So let $x\in H(\kappa).$ Define $X=\{x\}\cup \operatorname{trcl}(x).$ We have $|X| < \kappa$ by the definition of $H(\kappa).$
We have the following two theorems:
- (VI 3.9 pg 172) There is a finite conjunction $\chi$ of axioms of ZF - P + V=L such that $$ \forall M[(\mbox{$M$ is transitive }\land \chi^M)\to M=L(o(M))]$$ where $o(M) = M\cap \mathbf{Ordinals}$
- (IV 7.11 pg 140) Let $\mathbf{Z}$ be any transitive class and $\phi$ any sentence. Then $$ \forall X\subseteq Z\;[\mbox{$X$ is transitive} \to \\\exists M\;[X\subseteq M \land (\mbox{$M$ is transitive})\land |M|\le \max(|X|,\omega)\land \phi^{\mathbf Z}\leftrightarrow \phi^M ]]$$
I apply the second theorem with the $\chi$ from the first theorem, $\mathbf Z = \mathbf V=\mathbf L.$ I use the fact that the $X$ defined above is transitive to get a transitive $M$ such that $X\subseteq M,$ $|M| = |X|,$ and $\chi^M\leftrightarrow \chi.$
We know $\chi$ holds since we are working in ZF + V=L and $\chi$ is a conjunction of those axioms. So $\chi^M$ holds. Then, using this and the fact that $M$ is transitive in the first theorem above, we have $M=L(o(M)),$ so $x\in X\subseteq L(o(M))$ which means $x\in L(o(M)).$ Lastly, $|M| = |X| < \kappa,$ so $o(M) < |X|^+ \le \kappa,$ and thus $x\in L(o(M))\subseteq L(\kappa).\; \Box$
My question is whether this proof is correct or I've missed something. I largely did it by modifying the proof of Thm 4.6 that under V = L, $\mathcal P(L(\alpha))\subseteq L(\alpha^+),$ so it seems a little strange that he did the regular $\kappa$ case separately when this argument was available. Between that and the fact that the proof for arbitrary cardinal $\kappa$ in the newer edition is more complicated even though it seems to be, at its core, a similar reflection argument.
Edit. Looking at the argument in the 2013 edition a little more closely I see that there is actually a quite easy extension from regular cardinals to all cardinals. If it's true for every regular cardinal, it is true for every successor cardinal, so then we have $H(\kappa) = \bigcup_{\lambda<\kappa}H(\lambda^+)=\bigcup_{\lambda<\kappa}L(\lambda^+) = L(\kappa)$ for any limit cardinal $\kappa.$ So perhaps this is what he was going for. Still, the fact that his reflection argument separates the limit and successor cases gives me pause since I didn't need to do that, so still not confident I haven't missed something.