On the logical strength of the better quasi order with three elements

TL;DR

This paper explores the logical implications of the better quasi order property for a three-element order, establishing connections to significant subsystems of second-order arithmetic in reverse mathematics.

Contribution

It demonstrates that the better quasi order of a three-element set implies arithmetical recursion and transfinite recursion within reverse mathematics.

Findings

Arithmetical recursion follows from $ extbf{3}$ being $ extbf{BQO}$ over $ extbf{RCA}_0$.

Arithmetical transfinite recursion follows from $ extbf{3}$ being $ extbf{$ riangle^0_2$-$ extbf{BQO}$}$.

Establishes a link between combinatorial order properties and logical strength in reverse mathematics.

Abstract

The notion of better quasi order ($\mathsf{BQO}$), due to Nash-Williams, is very fruitful mathematically and intriguing from the standpoint of logic, due to several long-standing open problems. In the present paper, we make a significant step towards one of these: Let $\mathbf 3$ be the discrete order with three elements. We show that arithmetical recursion along the natural numbers ($\mathsf{ACA}_0^+$) follows from $\mathbf 3$ being $\mathsf{BQO}$, over the base theory $\mathsf{RCA_0}$ from reverse mathematics. Also over the latter, we deduce arithmetical transfinite recursion ($\mathsf{ATR}_0$) from the assumption that $\mathbf 3$ is $\Delta^0_2\text{-}\mathsf{BQO}$, which plays a role in work of Montalb\'an.