Variants of the chain-antichain principle in reverse mathematics

TL;DR

This paper investigates variants of the chain-antichain principle in reverse mathematics, using computability theory to distinguish between the general and restricted forms, including stable versions, revealing subtle differences.

Contribution

It introduces computability-theoretic reductions to differentiate the chain-antichain principle and its restriction, clarifying their logical and computational distinctions.

Findings

Distinct computational properties of CAC and its restriction

Stable versions exhibit different behaviors from non-stable ones

Formal reductions highlight subtle differences 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.