Weihrauch reducibility between Ramsey-type theorems and well-ordering principles at the level of $\Sigma^0_2$-induction: A pilot study

TL;DR

This paper explores the Weihrauch reducibility relations among Ramsey-type theorems and well-ordering principles at the $oldsymbol{ ext{Sigma}}^0_2$-induction level, revealing their complex interdependencies and incomparabilities.

Contribution

It establishes new Weihrauch-equivalence and incomparability results between well-ordering preservation principles and Ramsey theorems at the $oldsymbol{ ext{Sigma}}^0_2$-induction level.

Findings

Ordered Ramsey Theorem is Weihrauch-equivalent to the parallel product of well-ordering preservation and the Eventually Constant Tail principle.

The principle related to the jump of the Limited Principle of Omniscience is Weihrauch-incomparable with the well-ordering preservation principle.

The study clarifies the complex reducibility relations among these principles in reverse mathematics.

Abstract

We study the relations under Weihrauch reducibility of the well-ordering preservation principle for the operator $X \mapsto X^\omega$ and the Ordered Ramsey Theorem. Both principles are known to be equivalent to $\Sigma^0_2$-induction in Reverse Mathematics. We show that the Ordered Ramsey Theorem is Weihrauch-equivalent to the parallel product of the well-ordering preservation principle for the operator $X \mapsto X^\omega$ and the Eventually Constant Tail principle. By previous work from Pauly, Pradic and Sold\`a, the Ordered Ramsey Theorem is known to be Weihrauch-equivalent to the parallel product of the Eventually Constant Tail principle and the parallelization of the jump of the Limited Principle of Omniscience. We show that the latter pinciple and the well-ordering preservation principle for $X \mapsto X^\omega$ are Weihrauch-incomparable.