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

Abstract

The notion of better quasi order ( B Q O \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 3 \mathbf 3 be the antichain with three elements. We show that arithmetical recursion along the natural numbers ( A C A 0 + \mathsf {ACA}_0^+ ) follows from 3 \mathbf 3 being B Q O \mathsf {BQO} , over the base theory R C A 0 \mathsf {RCA_0} from reverse mathematics. Also over the latter, we deduce arithmetical transfinite recursion ( A T R 0 \mathsf {ATR}_0 ) from the assumption that 3 \mathbf 3 is Δ 2 0 - B Q O \Delta ^0_2\text {-}\mathsf {BQO} , which plays a role in work of Montalbán.