The Reverse Mathematics of CAC for trees

TL;DR

This paper investigates the reverse mathematical strength of the CAC for trees theorem, showing it is computationally weak and admits probabilistic solutions, with multiple characterizations aligning with existing literature.

Contribution

It establishes the robustness of TAC for trees, connects it with known theorems like TCAC and SHER, and demonstrates its computational weakness and probabilistic solvability.

Findings

CAC for trees admits probabilistic solutions

TAC for trees has multiple equivalent characterizations

The theorem is computationally weak

Abstract

CAC for trees is the statement asserting that any infinite subtree of $\mathbb{N}^{<\mathbb{N}}$ has an infinite path or an infinite antichain. In this paper, we study the computational strength of this theorem from a reverse mathematical viewpoint. We prove that TAC for trees is robust, that is, there exist several characterizations, some of which already appear in the literature, namely, the tree antichain theorem (TCAC) introduced by Conidis, and the statement SHER introduced by Dorais et al. We show that CAC for trees is computationally very weak, in that it admits probabilistic solutions.