Question

In the development of constructible sets via the Gödel operations, one of the main point seems to be that for $\Delta_0$ formulas, one can construct the sets of of elements verifying them with a finite number of Gödel operations, called $G_1$,...,$G_{10}$.

My questions are : does this means that Set theory with separation restricted to $\Delta_0$ formulas is finitely axiomatizable since a set is closed under a finite number of these operations iff it is closed under $\Delta_0$ formulas? Also since $\Delta_0$ formulas are absolute in transitive models, does this mean that if we consider $Th(M)$ with $M$ the fragment generated only by $\Delta_0$ formulas, then it is finitely axiomatizable? Now if you had $\Delta_1$ formulas is it still finitely axiomatizable? Finally, does this imply that a reflection theorem can't hold in Set Theory with $\Delta_0$ separation?

I hope my questions were accurate.

Answer 1

You might be interested in looking at Kripke-Platek Set Theory.

Yes, closure under the Gödel operations is equivalent to Δ₀-comprehension (plus union, pairing, and cartesian products). Over this base theory, Σ₁-reflection is equivalent to Δ₀-collection. Note that Σ₁-reflection does not prevent finite axiomatizability when the axioms are not Σ₁.