I've seen the "standard" proof that for regular $\mu>\omega$, the club filter on $\mu$ is normal—that is, closed under diagonal intersections. However, I was told about a potential alternate proof that involves elementary substructures, which seems interesting to me, although I can't figure out how to do it.
A potential formulation of the problem is as follows:
Let $\mu$ be an uncountable regular cardinal, and for each $\alpha<\mu$, suppose $C_\alpha$ is a club subset of $\kappa$. Let $X \preccurlyeq H(\mu^+)$ such that $|X|<\mu$ and $\langle C_\alpha:\alpha<\mu\rangle \in X$.
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Show that we can pick such an $X$ with $X\cap \mu$ transitive, and as such we can say $X\cap \mu = \delta_X$ for some ordinal $\delta_X<\mu $.
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Letting $\delta_X$ as in (1), show that $\delta_X \in \triangle_{\alpha<\mu} C_\alpha$
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Show that, in fact, we can find arbitrarily large such $\delta_X$, and thus $\triangle_{\alpha<\mu} C_\alpha$ is unbounded.
Combining (3) with the standard proof that $\triangle_{\alpha<\mu} C_\alpha$ is closed, we deduce $\triangle_{\alpha<\mu} C_\alpha$ is club.
Unfortunately, I'm very weak with these sorts of arguments and have not been able to make much progress (my motivation for working on this problem is precisely to get better at this).
For (1), my general idea is to start with some $X_0 \preccurlyeq H(\mu^+)$ with the above assumptions that may or may not be transitive. Then define $\beta_0=\sup(X\cap \mu)$, which is less than $\mu$ by regularity. Construct $X_1 \preccurlyeq H(\mu^+)$ with $X_0\subseteq X_1$ and $\beta_0\subseteq X_1$, still satisfying the same assumptions. Recursively construct $X_2,X_3$,etc in this manner, building up an $\omega$-length chain of elementary substructures of $H(\mu^+)$ where each "fills in the gaps" of the previous. Setting $X=\bigcup_{n\in \omega} X_n$, ideally we have arrived at a substructure of the desired form.
I know there is a lemma about unions of chains of elementary substructures themselves being elementary substructures, so I think this should work, but I'm not completely solid on its hypotheses so I would appreciate some help arguing exactly that $X$ is, in fact, an elementary substructure of $H(\mu^+)$ (or if it's not based on my construction alone, how to modify the construction such that it is).
As for $X\cap \mu$ being transitive, ideally this just follows from the "gap filling" at each step. If there were to be some gap in $X\cap \mu$, then this gap is witnessed for some finite $X_n$, and then gets filled in for $X_{n+1}$. Does this seem reasonable?
For (2), I'm not really sure where to begin. Obviously the elementary substructure containing the sequence of $C_\alpha$'s is important, but I'm not sure exactly why. I'd imagine I have to invoke elementarity and strategically pick some formula that will magically give me the result, but I'm not seeing what such a formula should be. Could I get some help here?
Finally for (3), it seems like given some $\beta < \mu$, we just add the additional requirement that $\beta\in X$, so the corresponding $\delta_X$ will necessarily be greater than $\beta$. Does this sound reasonable, or is there more to it?
Best Answer
(1) Your idea is fine.
Recall the lemma when dealing with chains like this:
It quickly follows $X\preceq H_{\mu^+}$.
Transitivity of $X\cap \mu$ is by construction.
(2) It suffices to show for all $\alpha<\delta$ that $\delta\in C_\alpha$. By $\alpha\in \delta\subseteq X$, $C_\alpha\in X$. So $$H_{\mu^+}\vDash\text{$C_\alpha$ is unbounded in $\mu$}$$ and so does $X$. It follows $C_\alpha\cap X$ is unbounded in $\mu\cap X$ so since $\delta<\mu$, $$\delta=\sup(\mu\cap X)=\sup(C_\alpha\cap X)\in C_\alpha$$
(3) Your idea works.