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.