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?
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!