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The Logic of Cardinality Comparison Without the Axiom of Choice

Matthew Harrison-Trainor, Dhruv Kulshreshtha

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.

The Logic of Cardinality Comparison Without the Axiom of Choice

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-). 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.
Paper Structure (13 sections, 22 theorems, 44 equations)

This paper contains 13 sections, 22 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Cardinal arithmetic without the axiom of choice
  3. Dedekind-finite sets and generalized finite cancellation
  4. Dedekind-infinite sets and covered generalized finite cancellation
  5. Formal language, models, and axioms
  6. Language
  7. Cardinality Models
  8. Logic for Finite Sets
  9. Logic for Dedekind-Finite Sets
  10. Logic for Arbitrary Sets
  11. Probability measures models for imprecise probability
  12. Urelements and Permutation Models
  13. Representation Theorems

Key Result

Theorem 1.1

Let $(\mathcal{P},\preceq)$ be a partial preorder. Then there is a model $\mathcal{U}$ of $\mathsf{ZF}$, and sets $(A_p)_{p \in \mathcal{P}}$ in $\mathcal{U}$, such that

Theorems & Definitions (57)

  • Theorem 1.1: Jech Jech66 and Takahashi Takahashi67; see Theorem 11.1 of Jech73
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4: Generalized Finite Cancellation
  • Theorem 1.5: Covered Generalized Finite Cancellation
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 2.1: Cantor-Schröder-Bernstein Theorem
  • Theorem 2.2: Division by $m$, Lindenbaum (unpublished), Tarski Tarski49
  • Theorem 2.3: Subtraction, see DC94
  • ...and 47 more