# What is the strength of adding limitation of size and a simple version of reflection to Ackermann set theory?

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 +

1. Extensionality: $$\forall x (x \in a \leftrightarrow x \in b) \to a=b$$

2. 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.

3. 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

1. Super-transitive: $$x \in V \wedge y \subset x \to y \in V$$

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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:

1. 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.

Let me denote as $$\mathsf{K}(V)$$ your system 1.+2.+3.+Super Transitivity. And as $$\mathsf{K}^{+}(V)$$ your system 1.+2.+3.+Limitation of size.
Note that the well-founded part translation gives an interpretation of $$\mathsf{K}(V)+\mathsf{Foundation}$$ in $$\mathsf{K}(V)$$ and $$\mathsf{K}^+(V)+\mathsf{Foundation}$$ in $$\mathsf{K}^+(V)$$. Hence we equivalently could calculate the consistency strength of the versions of the systems with Foundation.
The theory $$\mathsf{K}^{+}(V)+\mathsf{Foundation}$$ proves the same pure set-theoretic sentences as the theory $$\mathsf{ZF}+\mathsf{M}$$, where the scheme $$\mathsf{M}$$ is "any first-order definable club on the class $$On$$ contains a strongly inaccessible". Formally, $$\mathsf{M}$$ is the scheme $$\forall \alpha\in On \exists \beta>\alpha\; \varphi(\beta)\land \forall \alpha\in On(\forall \beta<\alpha\exists \gamma\in[\beta,\alpha)\; \varphi(\gamma)\to \varphi(\alpha))\to \exists \kappa(\varphi(\kappa)\land \mbox{\kappa is strongly inaccessible''}).$$ The intended models of the theory $$\mathsf{ZF}+\mathsf{M}$$ are $$\mathsf{V}_{\kappa}$$, where $$\kappa$$ is a Mahlo cardinal (in the same way the intended models of $$\mathsf{ZF}$$ are $$\mathsf{V}_{\kappa}$$, where $$\kappa$$ is strongly inaccessible).
From Fiedman we know that $$\mathsf{K}(V)+\mathsf{Foundation}$$ proves all the axioms of $$\mathsf{ZF}$$. Let us prove that $$\mathsf{K}^{+}(V)+\mathsf{Foundation}$$ proves all the instances of $$\mathsf{M}$$ (in the language without $$V$$). By reflection principle it is enough to prove in $$\mathsf{K}^{+}(V)+\mathsf{Foundation}$$ that $$\mathsf{M}$$ holds only for classes $$C$$ that are given by a pure set-theoretic formula with parameters from $$V$$. Note that for any pure set-theoretic formula $$\varphi(\vec{x})$$ the theory $$\mathsf{K}(V)\vdash\vec{x}\in V\to (\varphi^V(\vec{x})\leftrightarrow\varphi(\vec{x}))$$. Thus $$C$$ is unbounded in $$V\cap On$$. Hence $$(V\cap On)\in C$$. Therefore, $$C$$ contains an inaccessible.
Recall the standard fact that for any finite family of first-order formulas $$\varphi_i(\vec{x}_i)$$ the theory $$\mathsf{ZF}$$ proves that there exists $$\alpha$$ such that $$V_{\alpha}$$ reflects all $$\varphi_i$$, e.g. $$\forall \vec{x}_i\in V_{\alpha}\;(\varphi_i(\vec{x}_i)\leftrightarrow \varphi_i^{V_\alpha}(\vec{x}_i))$$. Observe that moreover for any finite family of first-order formulas closed under subformulas $$\mathsf{ZF}$$ proves that the class of all $$\alpha$$ s.t. $$V_{\alpha}$$ reflects all the formulas from the family is a club. Hence $$\mathsf{ZF}+\mathsf{M}$$ proves that any finite family of formulas is reflected on an inaccessible cardinal. Now assume $$\mathsf{K}^{+}(V)+\mathsf{Foundation}$$ proves some pure set-theoretic sentence $$\psi$$. Let consider all the family of all $$\varphi_i(\vec{x})$$ from the instances of reflection used in the proof. To prove $$\psi$$ in $$\mathsf{ZF}+\mathsf{M}$$ we interpret $$V$$ as an inaccessible cardinal that reflects all $$\varphi_i$$'s.