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On countability and representations

Sam Sanders

TL;DR

The paper develops a unified, higher-order Reverse Mathematics framework to analyze countability as a relation dependent on representations via an equivalence $=_{X}$. By formalizing countability with injections $Y:X\to \mathbb{N}$ and varying the ambient equality (e.g., $=_\mathbb{R}$, $=_{X}$), it derives strong equivalences: basic countability principles imply Feferman’s projection principle (BOOT) and countable choice ($\text{QF-AC}^{0,1}$). Across sections, classical results of Cantor, Sierpiński, and König are shown to yield powerful axioms when recast in this higher-order setting, including GS-type theorems and supremum principles for CRSC-spaces. The work highlights how seemingly modest notions of countability can encode substantial logical strength, linking topology, metric space theory, and order theory to foundational axioms in higher-order arithmetic. This provides a deeper understanding of the dependence of mathematical strength on representations and equivalence relations.

Abstract

The topic of this paper is the subtle interplay between countability and representations. In particular, we establish that the definition of countability of a certain set $X$ crucially hinges on the associated equivalence relation $=_{X}$. Armed with this knowledge, we study well-known and basic principles about countable sets, going back to Cantor, Sierpiński, and König, working in Kohlenbach's higher-order Reverse Mathematics. While these principles are relatively weak in second-order Reverse Mathematics, we obtain equivalences involving countable choice and Feferman's projection principle. The latter are essentially the strongest axioms studied in higher-order Reverse Mathematics and usually only come to the fore when dealing with the uncountable.

On countability and representations

TL;DR

The paper develops a unified, higher-order Reverse Mathematics framework to analyze countability as a relation dependent on representations via an equivalence . By formalizing countability with injections and varying the ambient equality (e.g., , ), it derives strong equivalences: basic countability principles imply Feferman’s projection principle (BOOT) and countable choice (). Across sections, classical results of Cantor, Sierpiński, and König are shown to yield powerful axioms when recast in this higher-order setting, including GS-type theorems and supremum principles for CRSC-spaces. The work highlights how seemingly modest notions of countability can encode substantial logical strength, linking topology, metric space theory, and order theory to foundational axioms in higher-order arithmetic. This provides a deeper understanding of the dependence of mathematical strength on representations and equivalence relations.

Abstract

The topic of this paper is the subtle interplay between countability and representations. In particular, we establish that the definition of countability of a certain set crucially hinges on the associated equivalence relation . Armed with this knowledge, we study well-known and basic principles about countable sets, going back to Cantor, Sierpiński, and König, working in Kohlenbach's higher-order Reverse Mathematics. While these principles are relatively weak in second-order Reverse Mathematics, we obtain equivalences involving countable choice and Feferman's projection principle. The latter are essentially the strongest axioms studied in higher-order Reverse Mathematics and usually only come to the fore when dealing with the uncountable.
Paper Structure (18 sections, 10 theorems, 29 equations)

This paper contains 18 sections, 10 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Aim and motivation
  3. Preliminaries
  4. Introduction
  5. Axioms of higher-order arithmetic
  6. Spaces, metric and otherwise
  7. Countability and representations
  8. Introduction
  9. Countability by any other name
  10. Countable linear orders
  11. Countable metric spaces
  12. Countable second-countable spaces
  13. Introduction
  14. Supremum principle
  15. The Ginsburg-Sands theorem
  16. ...and 3 more sections

Key Result

Theorem 2.3

The following are equivalent.

Theorems & Definitions (36)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.7: Hunter's conservation results
  • Definition 1.9
  • Definition 1.10
  • Definition 2.2
  • Theorem 2.3: $\textup{ACA}_{0}^{\omega}+\textup{QF-AC}^{0,1}$
  • proof
  • Corollary 2.4: $\textup{ACA}_{0}^{\omega}$
  • proof
  • ...and 26 more