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Variants of the chain-antichain principle in reverse mathematics

Noah A. Hughes

TL;DR

The paper analyzes variants of the chain-antichain principle in reverse mathematics, focusing on restricting $CAC$ to $ ext{omega}$-ordered posets and on stable versions. It develops a robust computability-theoretic framework including instance-solution problems and several reductions, and proves that $CAC$ and $ ext{omega-}{ m CAC}$ are equivalent under generalized Weihrauch reductions, while they are not computably equivalent. It further shows that ${ m SCAC}$ is computably equivalent to $ ext{omega-}{ m SCAC}$ and exhibits a uniform Weihrauch profile for stable variants, but establishes strong separations between ${ m SCAC}$ and $ ext{omega-}{ m SCAC}$ via forcing and tree-labeling techniques. The results highlight subtle distinctions between RM principles when viewed through computability-theoretic reductions and demonstrate the power of forcing methods and tree-labeling in separating stable variants, with implications for the fine structure of poset-based combinatorial principles in reverse mathematics.

Abstract

Restricting the chain-antichain principle CAC to partially ordered sets which respect the natural ordering of the integers is a trivial distinction in the sense of classical reverse mathematics. We utilize computability-theoretic reductions to formally establish distinguishing characteristics of CAC and the aforementioned restriction to elaborate on the apparent differences obfuscated over RCA0. Stable versions of both principles are also analyzed in this way.

Variants of the chain-antichain principle in reverse mathematics

TL;DR

The paper analyzes variants of the chain-antichain principle in reverse mathematics, focusing on restricting to -ordered posets and on stable versions. It develops a robust computability-theoretic framework including instance-solution problems and several reductions, and proves that and are equivalent under generalized Weihrauch reductions, while they are not computably equivalent. It further shows that is computably equivalent to and exhibits a uniform Weihrauch profile for stable variants, but establishes strong separations between and via forcing and tree-labeling techniques. The results highlight subtle distinctions between RM principles when viewed through computability-theoretic reductions and demonstrate the power of forcing methods and tree-labeling in separating stable variants, with implications for the fine structure of poset-based combinatorial principles in reverse mathematics.

Abstract

Restricting the chain-antichain principle CAC to partially ordered sets which respect the natural ordering of the integers is a trivial distinction in the sense of classical reverse mathematics. We utilize computability-theoretic reductions to formally establish distinguishing characteristics of CAC and the aforementioned restriction to elaborate on the apparent differences obfuscated over RCA0. Stable versions of both principles are also analyzed in this way.
Paper Structure (12 sections, 22 theorems, 21 equations, 1 figure)

This paper contains 12 sections, 22 theorems, 21 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Reverse Mathematics
  3. Computability-theoretic reductions
  4. Definitions
  5. Applications of the reverse mathematical proof
  6. Computable reductions
  7. Uniformity and stability
  8. Separating the stable versions
  9. Separation over Weihrauch reducibility
  10. Separation over strong computable reducibility
  11. Tree labeling
  12. Proving the result

Key Result

Theorem 2.1

There is an infinite computable poset $(H, \leqslant_H)$ with $H \subseteq \omega$ that does not contain a $\Delta^{0}_{2}$ infinite chain or a $\Delta^{0}_{2}$ infinite antichain.

Figures (1)

  • Figure 1: The left diagram summarizes the implications between each sort of reduction. The right diagram provides a graphical representation of the reduction paradigm in the specific context of a strong Weihrauch reduction.

Theorems & Definitions (63)

  • Definition 1.1
  • Theorem 2.1: Theorem 3.1 of herrmann_infinite_2001
  • Theorem 2.2
  • proof
  • Remark 1
  • Definition 2.1
  • Remark 2
  • Corollary 2.3
  • proof
  • Remark 3
  • ...and 53 more