The predicative Frege hierarchy

Abstract

In this paper, we characterize the strength of the predicative Frege hierarchy, Pn+1V, introduced by John Burgess in his book [Burg05]. We show that Pn+1V and Q + conn(Q) are mutually interpretable. It follows that PV := P1V 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 P2V is mutually interpretable with Kalmar Arithmetic (a.k.a. EA, EFA, IΔ0+EXP, Q3). The fact that P2V interprets EA, was proved earlier by Burgess. We provide a different proof. Each of the theories Pn+1V is finitely axiomatizable. Our main result implies that the whole hierarchy taken together, PωV, is not finitely axiomatizable. What is more: no theory that is mutually locally interpretable with PωV is finitely axiomatizable.