The Logic of Cardinality Comparison Without the Axiom of Choice

TL;DR

This paper develops a complete logical framework for comparing the sizes of sets within ZF set theory without the Axiom of Choice, focusing on Boolean operations and cardinality relations, including special cases like Dedekind-finite sets.

Contribution

It provides a complete axiomatization for reasoning about set cardinalities without the Axiom of Choice, introducing principles like Generalized Finite Cancellation.

Findings

A complete axiomatization for cardinality comparison in ZF

Identification of principles needed for Dedekind-finite sets

Connection to imprecise probability comparison principles

Abstract

We work in the setting of Zermelo-Fraenkel set theory without assuming the Axiom of Choice. We consider sets with the Boolean operations together with the additional structure of comparing cardinality (in the Cantorian sense of injections). What principles does one need to add to the laws of Boolean algebra to reason not only about intersection, union, and complementation of sets, but also about the relative size of sets? We give a complete axiomatization. A particularly interesting case is when one restricts to the Dedekind-finite sets. In this case, one needs exactly the same principles as for reasoning about imprecise probability comparisons, the central principle being Generalized Finite Cancellation (which includes, as a special case, division-by-$m$). In the general case, the central principle is a restricted version of Generalized Finite Cancellation within Archimedean classes which we call Covered Generalized Finite Cancellation.