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 P V : = 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 Δ 0 + EXP , Q3). The fact that P 2 V interprets EA was proved earlier by Burgess. We provide a different proof. f the theories P n + 1 V is finitely axiomatizable. Our main result implies that the whole hierarchy taken together, P ω V , is not finitely axiomatizable. What is more: no theory that is mutually locally interpretable with P ω V is finitely axiomatizable.