I'm going through the proof that if $\kappa$ is inaccessible then $V_\kappa \vDash \mathrm{ZFC}$ and how thus we have $\mathrm{ZF} \nvdash \text{"There exist inaccessible cardinals"}$.

So the last part is by taking $\kappa$ to be the *least* inaccessible cardinal and then showing that $V_\kappa \vDash \text{"There is no inaccessible cardinal"}$.

But for this to work, we must show that being an inaccessible cardinal is absolute for $V_\kappa$, yes? Otherwise, it is possible $V_\kappa$ could just think it has inaccessible cardinals even though it doesn't.

So how do we prove that being an inaccessible cardinlas is absolute? Jech simply leaves it to the reader, and I'm struggling to find it elsewhere. It must be easy, because no one even seems to bother proving it, or perhaps it's not necessary at all? Am I missing something obvious? Is it not neccessary for this notion to be absolute?

`$V_\kappa$`

to get $V_\kappa$. $\endgroup$3more comments