# Natural models of Ackermann Set Theory

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:

1. Is there a classical large cardinal axiom P such that ZFC+P proves that AST has a natural model?
2. 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?