Misunderstanding Löwenheim–Skolem

Question

The Löwenheim–Skolem theorem shows that we can find a countable elementary submodel of $V$ that satisfies $ZFC$. [assuming, Con$(ZFC$)]. Call this set $U$. Then by the definition of elementary submodel, $V$ and $U$ must believe the same formulae. Let $\kappa$ be a cardinal in $U$ that $U$ believes to be uncountable. (Such a cardinal must exist as $V$ believes that there are uncountable cardinals, therefore so does $U$). Then as $U$ countable, $\kappa$ must be countable (as seen from $V$). However, now $U$ and $V$ disagree about the formula '$\kappa$ is uncountable', which seems (to me) to contradict the definition of elementary submodel. Where have I gone wrong here?

Answer 1

Good question! This is a subtle point. The error is when you write:

$(*)$ Then as $U$ countable, $\kappa$ must be countable (as seen from $V$).

This is not the case! Presumably, the reason for believing $(*)$ is (something like) "$\kappa$ in $U$, so $\kappa\subseteq U$," but this assumes that $U$ is transitive. (A set $A$ is transitive if $y\in x\in A\implies y\in A$.)

This need not be the case; in fact, your exact argument shows that $U$ is never transitive! Rather, all we can conclude from the countability of $U$ is that the set $$\kappa\cap U$$ must be countable (as seen from $V$). Basically, $U$ will contain lots of elements which are uncountable sets, but $U$ will contain only "countably much" of each.