In attempting to accommodate (S1)–(S3) simultaneously, we must confront a tension which has led many authors to follow Dummett (1975) in concluding that strict finitism lacks a coherent formulation. For suppose we let \(F(x)\) symbolize that \(x\) denotes a feasible number in Yessenin-Volpin’s sense. Then it would seem to follow from (S1) that we ought to accept