Constructing Sequences One Step at a Time

TL;DR

The paper introduces a novel method for constructing Turing ideals to explore the logical strength of principles in reverse mathematics, leading to new separations and insights under Weak König's Lemma.

Contribution

It presents a new construction technique for Turing ideals that enables proving separations between principles in reverse mathematics below CAC, especially with WKL.

Findings

CAC+WKL does not imply the thin set theorem for pairs

The product of well-quasi-orders is strictly between CAC and ADS in strength

New separations in reverse mathematics principles under WKL

Abstract

We propose a new method for constructing Turing ideals satisfying principles of reverse mathematics below the Chain-Antichain Principle (CAC). Using this method, we are able to prove several new separations in the presence of Weak Konig's Lemma (WKL), including showing that CAC+WKL does not imply the thin set theorem for pairs, and that the principle "the product of well-quasi-orders is a well-quasi-order" is strictly between CAC and the Ascending/Descending Sequences principle, even in the presence of WKL.