Finite sets, mappings, cardinals, and arithmetic in intuitionistic New Foundations

Abstract

$\mathsf{NF}$ set theory using intuitionistic logic is called $i\mathsf{NF}$. We develop the theories of finite sets and their power sets and mappings, finite cardinals and their ordering, cardinal exponentiation, addition, and multiplication. We follow Rosser and Specker with appropriate constructive modifications, especially replacing "arbitrary subset" by "separable subset" in the definitions of exponentiation and order. It is not known whether $i\mathsf{NF}$ proves that the set of finite cardinals is infinite, so the whole development must allow for the possibility that there is a maximum integer; arithmetical computations might "overflow" as in a computer or odometer, and theorems about them must be carefully stated to allow for this possibility. The work presented here is intended as a basis for further investigations of $i\mathsf{NF}$, including the development of Bishop-style constructive mathematics in $i\mathsf{NF}$.