The predicative Frege hierarchy

In this paper, we characterize the strength of the predicative Frege hierarchy, P(n+1)V, introduced by John Burgess in his book [J. Burgess, Fixing frege, in: Princeton Monographs in Philosophy, Princeton University Press, Princeton, 2005]. We show that p(n+1)V and Q+con(n)(Q) are mutually interpretable. It follows that PV:= P(1)V is mutually interpretable with Q. This fact was proved earlier by Mihai Ganea in [M. Ganea, Burgess' PV is Robinson's Q. The journal of Symbolic Logic 72 (2) (2007) 619-624] using a different proof. Another consequence of the our main result is that P(2)V is mutually interpretable with Kalmar Arithmetic (a.k.a. EA, EFA, I Delta(0) + EXP, Q3). The fact that P(2)V interprets EA was proved earlier by Burgess. We provide a different proof. Each of the theories p(n+1)V is finitely axiomatizable. Our main result implies that the whole hierarchy taken together, P(omega)V, is not finitely axiornatizable. What is more: no theory that is mutually locally interpretable with P(omega)V is finitely axiomatizable. (C) 2009 Elsevier B.V. All rights reserved.