A Note on Finiteness in the Predicative Foundations of Arithmetic

Abstract

Recently, Feferman and Hellman (and Aczel) showed how to establish the existence and categoricity of a natural number system by predicative means given the primitive notion of a finite set of individuals and given also a suitable pairing function operating on individuals. This short paper shows that this existence and categoricity result does not rely (even indirectly) on finite-set induction, thereby sustaining Feferman and Hellman's point in favor of the view that natural number induction can be derived from a very weak fragment of finite-set theory, so weak that finite-set induction is not assumed. Many basic features of finiteness fail to hold in these weak fragments, conspicuously the principle that finite sets are in one-one correspondence with a proper initial segments of a (any) natural number structure. In the last part of the paper, we propose two prima facie evident principles for finite sets that, when added to these fragments, entail this principle.