Lets denote any set that is "the set of all strictly smaller subsets of it" as size- unreachable. Formally: $$\operatorname {size-unreachable}(X) \iff \\\forall Y (Y \in X \iff Y \subset X \land |Y|<|X|)$$Now $V_\omega$ qualifies for such a set.
I gave this definition in a prior posting, where I asked about examples of such sets, the answer given is an $H_\kappa$ set where $|H_\kappa|=\kappa$, (where the $H$ function is defined as: $H_\kappa=\{x:|x|<\kappa\land \forall y\in \operatorname{trcl}(x)\; |y|<\kappa\} $).
My question is about the equivalence of the following two theories:
- ZFC + every set is an element of some size-unreachable $V_\kappa$ stage.
- Extensionality, Separation, and Unreachable Reflection.
Unreachable Reflection: if $\phi$ is a formula (defined terms allowed) that doesn't use $V_\kappa$, and $\kappa$ is not free, then: $$\phi \implies \exists \kappa : \operatorname {size-unreachable}(V_\kappa) \land V_\kappa \models \phi$$
