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 -comprehension and of Heck’s ramified predicative second-order system.
"The Finitistic Consistency of Heck’s Predicative Fregean System." Notre Dame J. Formal Logic 56 (1) 61 - 79, 2015. https://doi.org/10.1215/00294527-2835110