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.



Finite axiomatizability and constructible sets
You might be interested in looking at [Kripke-Platek Set Theory](http://en.wikipedia.org/wiki/Kripke%E2%80%93Platek_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.


