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Let $\kappa$ be any regular cardinal, and let $<C_i\mid i<\kappa>$ be a sequence of club sets. Define their diagonal intersection $\Delta C_i$ as follows:

$\Delta C_i = \{\alpha<\kappa \mid \forall i<\alpha:\alpha\in C_i\}$

I must prove that this set is also club. Now, I would like to consider the following supposed counterexample. Clearly, $\omega$ is regular, so if I define the following sequence of sets:

$C_i =\{p_i ^n \mid n\in\mathbb N\}$

Where $p_i$ is the $i$th prime number, we have a sequence of infinite (trivially) club sets with no pairwise intersection. If I understand correctly, this means that every number larger than $1$ is an element of $0$ or $1$ sets, and is not an element of the diagonal intersection. In particular, the diagonal intersection contains only $2$ elements, and so it is not unbounded and not club.

I assume this counterexample fails somewhere, as I was supposed to prove the claim. Can anyone please help me locate my error?

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    $\begingroup$ I looked at a few set theory books on my shelves (because I mainly know about the situation for $\kappa = {\omega}_1$ and wasn't sure of the exact hypothesis used for other cardinals), and the assumption for this diagonal intersection result is that $\kappa$ is a regular cardinal with uncountable cofinality. Perhaps you overlooked this aspect, or maybe what you're looking at overlooked saying this (or made this assumption earlier in the discussion and you didn't notice)? $\endgroup$ – Dave L. Renfro Dec 9 '18 at 17:22
  • $\begingroup$ Ok, so there really was some confusion. Thank you for the clarification! $\endgroup$ – Uri George Peterzil Dec 9 '18 at 17:29
  • $\begingroup$ This result (when $\kappa=cf (\kappa)\geq \omega_1$) is used in proving an important tool known as Fodor's Lemma, or the Pressing-Down Lemma. $\endgroup$ – DanielWainfleet Dec 9 '18 at 20:37
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When talking about clubs/stationary sets/etc., we have to restrict to uncountable regular cardinals - or at least, ordinals of uncountable cofinality - to get a nontrivial theory. For example, both $\{$evens$\}$ and $\{$odds$\}$ are club in $\omega$, but their intersection is empty; so it's not even diagonal intersection that fails, but regular (hehe) intersection!

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