I have a problem about elementary submodels of ZFC

Question

So suppose $\kappa$ inaccessible so that $V_\kappa$ is a model of ZFC, using Skolem and the Mostoswky collapse we have a countable elementary submodel $M$ of $V_\kappa$. This implies that for any formula $F(x)$ and any $a$ in $M$ we have : $V_\kappa$ verifies $F(a)$ iff $M$ verifies $F(a)$.

$\omega$ (the set of all integers) is absolute between transitive models so the formula "there is a bijective $f$ : $\omega$ into $A$" has only $A$ as a free variable.

Suppose $A$ is countable (and so countable in $V_\kappa$) we have $V_\kappa$ verifies : "there is a bijective $f$ : $\omega$ into $A$"

If $A$ is in $M$, by $M$ being an elementary submodel we have $M$ verifies : "there is a bijective $f$ : $\omega$ into $A$", so $A$ is countable in $M$, this implies that $M$ thinks that every (not finite) set is countable, which is false.

I can't find were the error is, I know that $M$ is supposed to not have such $f$ so that $M$ thinks that some sets are uncountable.

Answer 1

You have to distinguish between $A$ (which is in $M$ and might be uncountable) and $A\cap M$ (which is always countable but might not be in $M$).

Let's look at $A=\omega_1$ for concreteness. Since $\omega_1$ is definable without parameters, whenever $M\preccurlyeq V_\kappa$ we'll have $\omega_1\in M$. However, if $M$ is countable we will also have $\omega_1\not\subseteq M$ - this is fine, it just means that $M$ isn't transitive. The relevant thing which $V_\kappa$ sees is countable is $\omega_1\cap M$, but that's not an element of $M$. So there's nothing useful to apply elementarity to.

We could "transitivize" $M$ by taking the Mostowski collapse, but this would kill elementarity: if $M\preccurlyeq V_\kappa$ then there is a (unique) transitive $N$ such that $M\cong N$, but we don't in general have $N\preccurlyeq V_\kappa$, and in particular we never have this if $M$ is countable. So we wind up with a dichotomy: we can have countable elementary submodels which aren't transitive, or countable transitive submodels which aren't elementary.