TWO (OR THREE) NOTIONS OF FINITISM

Abstract

Finitism is given an interpretation based on two ideas about strings (sequences of symbols): a replacement principle extracted from Hilbert’s work and a counting principle inspired by Tait. These principles are used to justify an equational arithmetic ${\cal T}_{{\cal L}^2 } $ based on the algebra of lower elementary functions. The extension of this algebra to Grzegorczyk’s class ɛ2 can be justified by means of an additional finitistic choice principle, thus obtaining a second equational theory ${\cal T}_{{\cal E}^2 } $. It is unknown whether ${\cal T}_{{\cal E}^2 } $ is strictly stronger than ${\cal T}_{{\cal L}^2 } $ since ɛ2 may coincide with the class of lower elementary functions.If the objects of arithmetic are taken to be binary numerals instead of tally numerals, then it becomes possible to provide a finitistic justification for a theory ${\cal T}_B $ that may be incomparable to ${\cal T}_{{\cal E}^2 } $ (neither of the two includes the other). I conclude by suggesting that the equational theory of Kalmar elementary functions is a strict upper bound for finitistic arithmetic.