The following theory is formulated in first order predicate logic with extra-logical primitives of equality $``="$, membership $``\in"$, and a single primitive constant symbol $V$ denoting the class of all sets.
The axioms are those of first order identity theory +
Extensionality: $\forall x (x \in a \leftrightarrow x \in b) \to a=b$
Class comprehension: if $\varphi(y)$ is a formula in which the symbol $``y"$ occurs free, then all closures of: $\exists x \forall y (y \in x \leftrightarrow y \in V \wedge \varphi(y))$ are axioms.
Reflection: if $\varphi(y, x_1,..,x_n)$ is a formula in $FOL(=,\in)$, in which only $y,x_1,..,x_n$ occur free, then:
$$\forall x_1,..,x_n \in V \\ [\exists y (\varphi(y,x_1,..,x_n)) \to \exists y \in V (\varphi(y,x_1,..,x_n))]$$
is an axiom
- Super-transitive: $x \in V \wedge y \subset x \to y \in V$
This system would interpret the whole of Ackermann's set theory [Harvey Friedman]! Yet to me it looks more elegant than Ackermann's. However my question here is that if we replace the last axiom by a limitation of size axiom which states that $V$ is a class of all subsets of it that are strictly smaller than it in cardinality, i.e. formally this is:
- Limitation of Size: $\forall x (x \in V \leftrightarrow x \subset V \wedge |x|<|V| )$
then how much this would increase the consistency strength of this theory?
I mean this would increase the strength beyond $ZFC$ and $MK$, since $V$ would be inaccessible, and this is describable in a first order logic formula, and so by reflection there would exist a set in $V$ that is inaccessible.