The predicative Frege hierarchy

In this paper, we characterize the strength of the predicative Frege hierarchy, PV, introduced by John Burgess in his book [Bur05]. We show that PV and Q + con(Q) are mutually interpretable. It follows that PV := PV is mutually interpretable with Q. This fact was proved earlier by Mihai Ganea in [Gan06] using a different proof. Another consequence of the our main result is that PV is mutually interpretable with Kalmar Arithmetic (a.k.a. EA, EFA, I∆0+EXP, Q3). The fact that P V interprets EA, was proved earlier by Burgess. We provide a different proof. Each of the theories PV is finitely axiomatizable. Our main result implies that the whole hierarchy taken together, PV, is not finitely axiomatizable. What is more: no theory that is mutually locally interpretable with PV is finitely axiomatizable.