I'm trying to prove this theorem, but is it even true? A Tarski's class is a set $T$ such that:
- For each $y\in T$, ${\cal P}(y)\subseteq T$ and ${\cal P}(y)\in T$, and
- for each $A\subseteq T$ such that $A\prec T$ (a.k.a. $A$ is strictly smaller than $T$), $A\in T$.
Tarski's axiom says that for each set $x$, there is a Tarski's class $T\ni x$. I have managed to prove that $V_\kappa$ is a Tarski's class iff $\kappa$ is (strongly) inaccessible, but I don't think this is sufficient to prove that there is a proper class of inaccessibles, because I don't have any control over the form of the Tarski's classes that arise from the axiom. One reasonable way of wrangling the form of the Tarski's classes is to define the function $T(x)=\bigcap\{T\mid x\in T\wedge T\mbox{ is a Tarski's class}\}$, which is well-defined given Tarski's axiom and is a Tarski's class because Tarski's classes are closed under arbitrary intersection.
The only step that remains is to prove that $T(x)=V_\kappa$ where $\kappa$ is the least inaccessible with $\kappa>{\rm rank}(x)$, but I'm at a loss for how to approach the problem. Any tips?