Can we get rid of the primitive symbol $V$ in Ackermann's set theory without increment in consistency strength?

EDIT: Ackermann had presented his theory with a new primitive added to the language of set theory that is the "set-hood" primitive one place predicate symbol $$\mathcal M$$ or in common equivalent formulations the "class of all sets" primitive constant symbol $$V$$. However, the underlying intuition behind this addition is somewhat vague. And it appears to be technical rather than reflecting intuitive conception.

I was thinking of ways to get rid of this new primitive and yet still get a theory that more or less resemble Akermann's or an extension of Ackermann's set theory.

The nearest I got to is the theory exposited below.

However, this theory is much stronger than Ackermann and ZFC!

So I wonder if every attempt to get rid of this new primitive [i.e.; $$V$$ or $$\mathcal M$$] would result in a stronger theory than Ackermann's.

EXPOSITION:

Language: Mono-sorted first order predicate logic with primitive binary relations of equality $$="$$ and class membership $$\in"$$.

Axioms: ID axioms +

Extensionality: $$\forall z (z \in x \leftrightarrow z \in y) \to x=y$$

Class comprehension: if $$\phi$$ is a formula in which $$x$$ doesn't occur free, then all closures of $$\exists x \forall y (y \in x \leftrightarrow \exists z (y \in z) \wedge \phi)$$; are axioms.

We define $$transitive.inclusive(x)$$ to mean that $$x$$ is the class of all subsets of it of strictly smaller cardinality than it. Formally that is:

Define: $$transitive.inclusive(x) \iff \forall y (y \in x \leftrightarrow y \subset x \wedge |y| < |x|)$$

Reflection: for $$n=0,1,2,..$$; if $$\psi$$ is a formula that doesn't use the symbol $$W$$, having all and only symbols $$y,x_1,..,x_n$$ occurring free, then: $$\exists W \big{(} transitive.inclusive (W) \wedge \\ \forall x_1,..,x_n \in W [\forall y (\psi \to y \subset W) \to \forall y (\psi \to y \in W) ]\big{)}$$; is an axiom.

If we only require $$W$$ to be transitive in the reflection schema then we get to interpret Zermelo set theory.

• I tried editing to fix up some funky TeX, but there is a sentence fragment ("where a special formula …") that I couldn't figure out how to integrate. Is it meant to appear before the massive displayed formula? – LSpice Mar 25 at 20:28
• I think your latest edit renders the question too vague. – Noah Schweber Apr 7 at 22:56
• @NoahSchweber, Ok, I've re-edited it. I hope this version is clearer. – Zuhair Al-Johar Apr 12 at 19:51