The logical strength of minimal bad arrays

TL;DR

This paper explores the logical strength of the minimal bad array lemma in the theory of better quasi orders, revealing its equivalence to a significant set existence principle in reverse mathematics.

Contribution

It establishes that the minimal bad array lemma is equivalent to -comprehension over \u03a4, highlighting its logical strength.

Findings

Minimal bad array lemma is equivalent to -comprehension.

The lemma's strength is characterized within reverse mathematics.

It connects combinatorial principles to logical set existence axioms.

Abstract

This paper studies logical aspects of the notion of better quasi order, which has been introduced by C. Nash-Williams (Mathematical Proceedings of the Cambridge Philosophical Society 1965 & 1968). A central tool in the theory of better quasi orders is the minimal bad array lemma. We show that this lemma is exceptionally strong from the viewpoint of reverse mathematics, a framework from mathematical logic. Specifically, it is equivalent to the set existence principle of $\Pi^1_2$-comprehension, over the base theory $\mathsf{ATR_0}$.