The Definability Strength of Combinatorial Principles

TL;DR

This paper introduces the concept of definability strength for combinatorial principles, analyzing how certain Ramsey-type theorems influence the complexity of defining specific sets within reverse mathematics.

Contribution

It formalizes the notion of definability strength and applies it to Ramsey's Theorem, revealing which variants can simplify definitions of certain sets and deriving new implications in reverse mathematics.

Findings

Some consequences of Ramsey's Theorem simplify definitions of $ ext{Delta}^0_2$ sets.

Other consequences of Ramsey's Theorem do not aid in simplifying definitions.

Results lead to strengthened and new theorems in reverse mathematics.

Abstract

We introduce the definability strength of combinatorial principles. In terms of definability strength, a combinatorial principle is strong if solving a corresponding combinatorial problem could help in simplifying the definition of a definable set. We prove that some consequences of Ramsey's Theorem for colorings of pairs could help in simplifying the definitions of some $\Delta^0_2$ sets, while some others could not. We also investigate some consequences of Ramsey's Theorem for colorings of longer tuples. These results of definability strength have some interesting consequences in reverse mathematics, including strengthening of known theorems in a more uniform way and also new theorems.