Reverse Mathematics and initial intervals

TL;DR

This paper explores the reverse mathematics of Bonnet's theorems on partial orders, linking their proofs to subsystems like ACA_0, ATR_0, and WKL_0, and establishing equivalences with classical results.

Contribution

It establishes the logical strength of Bonnet's theorems within reverse mathematics, connecting their directions to specific subsystems and proving new equivalences.

Findings

Left to right directions are equivalent to ACA_0 and ATR_0.

Opposite directions are provable in WKL_0 but not in RCA_0.

Equivalence of a classical result with ACA_0 regarding strong antichains.

Abstract

In this paper we study the reverse mathematics of two theorems by Bonnet about partial orders. These results concern the structure and cardinality of the collection of the initial intervals. The first theorem states that a partial order has no infinite antichains if and only if its initial intervals are finite unions of ideals. The second one asserts that a countable partial order is scattered and does not contain infinite antichains if and only if it has countably many initial intervals. We show that the left to right directions of these theorems are equivalent to ACA_0 and ATR_0, respectively. On the other hand, the opposite directions are both provable in WKL_0, but not in RCA_0. We also prove the equivalence with ACA_0 of the following result of Erd\"os and Tarski: a partial order with no infinite strong antichains has no arbitrarily large finite strong antichains.