Consider Ackermann's Set Theory (as described here, §6, including the axiom of regularity for sets), henceforth denoted AST, as a theory in the language $\left\langle \in,\mathbf{V} \right\rangle$ where $\mathbf{V}$ is a constant symbol intended to represent the von Neumann universe in models of AST.
In Natural Models of Ackermann's Set Theory, Rudolf Grewe studies models of AST of the form $(V_{\alpha},\in,V_{\beta})$ for ordinals $\beta<\alpha$, so-called natural models of AST.
Grewe proves that for ordinals $\alpha>\beta$ where $\alpha$ is limit, the structure $(V_{\alpha},\in,V_{\beta})$ is a model of AST if and only if $V_{\beta}$ is not $\left\langle \in \right\rangle$-definable in $V_{\alpha}$ with parameters in $V_{\beta}$. Equivalently, the ordinal $\beta$ must not be definable in $V_{\alpha}$ with parameters in $V_{\beta}$.
This doesn't seem to be a very good thing to know if one wants to produce natural models of AST since the undefinability of an ordinal is a meta-statement. However this makes it easy to see what cannot be a model of AST.
Later in the article, Grewe proves that in fact $V_{\beta}$ must be a model of ZF. If $\kappa$ is a cardinal which is smallest to satisfy some large cardinal property $P$ which is a $\left\langle \in \right\rangle$-sentence, the ordinal $\kappa$ will always be definable in $V_{\alpha}$ without parameters by this very minimality. Therefore no such $V_{\kappa}$ is a candidate to interpret V in our natural model.
Of course the undefinability in $\left\langle \in \right\rangle$ of the interpretation of $\mathbf{V}$ in a model of AST is an important feature of the theory, but I find this prevalence in the case of natural models to be troubling.
I have two questions pertaining to this, one of which is vague:
- Is there a classical large cardinal axiom P such that ZFC+P proves that AST has a natural model?
- Is the importance of the undefinability in $\mathbf{V}$ of some rank $V_{\beta}$ in the von Neumann hierarchy a specific feature in AST, or is it a recurring theme in set-theory that has lied beyond my amateurish gaze for now?
