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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.

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  • $\begingroup$ I think Godel operation is just one way to materialize the idea of primitive recursive functions. It is amazing since Godel wrote this before the idea of primitive recursive functions came about. All are already at the back of Godel mind. $\endgroup$ – abcdxyz Apr 5 '10 at 2:18
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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.

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