# lo.logic – Can this reflexive class theory interpret ZFC?

Theory of reflective sets $$mathsf {RfST}$$ is formulated in a first-order predicate logic with extra-logical equality primitives $$“ = "$$membership $$“ in "$$, and a single constant primitive symbol $$V$$ designating the class of all sets.

Axioms are those of the theory of the first order identity +

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, so 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 for 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. Transitive: $$forall x in V forall y in x (y in V)$$

2. Foundation: $$exists m in x to exists y in x forall z in x (z not in y)$$

/ Definition of the completed theory.

Personally, I think this axiometry is the most elegant theory of sets / classes I know!

Can this theory interpret $$ZFC$$ more than $$L$$?

I speak clearly of the axioms of matching, union, separation and replacement $$V$$ pure formulas, the infinite, are all provable here, but the power is not easily provable here, but all together $$V$$ it's an element of a scene $$L _ { kappa}$$, would have all his subsets in $$L _ { kappa}$$ to be in $$V$$because they are all definable by well-defined formulas; we must therefore be able to reflect the existence of a set of all these subsets inside. $$V$$, thus interpreting $$ZFC + V = L$$.