The Finitistic Consistency of Heck’s Predicative Fregean System

Abstract

Frege’s theory is inconsistent (Russell’s paradox). However, the predicative version of Frege’s system is consistent. This was proved by Richard Heck in 1996 using a model-theoretic argument. In this paper, we give a finitistic proof of this consistency result. As a consequence, Heck’s predicative theory is rather weak (as was suspected). We also prove the finitistic consistency of the extension of Heck’s theory to ${\Delta }_1^1$-comprehension and of Heck’s ramified predicative second-order system.