Finite sets, mappings, cardinals, and arithmetic in intuitionistic NF

TL;DR

This paper develops the theory of finite sets, cardinals, and arithmetic within intuitionistic NF set theory, accounting for potential finiteness and constructiveness constraints, and sets the stage for further constructive mathematics in this framework.

Contribution

It introduces a constructive development of finite set theory, cardinal arithmetic, and orderings in intuitionistic NF, considering the possibility of finite maxima and overflow.

Findings

Finite set and power set theories adapted to intuitionistic logic.

Development of finite cardinal arithmetic with potential maximums.

Framework for further constructive mathematics in iNF.

Abstract

NF set theory using intuitionistic logic is called iNF. 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 iNF 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 substrate for further investigations of iNF, including the development of Bishop-style constructive mathematics in iNF.