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THE REVERSE MATHEMATICS OF
${\mathsf {CAC\ FOR\ TREES}}$
- Part of
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- Journal:
- The Journal of Symbolic Logic / Volume 89 / Issue 3 / September 2024
- Published online by Cambridge University Press:
- 28 April 2023, pp. 1189-1211
- Print publication:
- September 2024
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- Article
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${\mathsf {CAC\ for\ trees}}$ 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 
${\mathsf {CAC\ for\ trees}}$ is robust, that is, there exist several characterizations, some of which already appear in the literature, namely, the statement 
$\mathsf {SHER}$ introduced by Dorais et al. [8], and the statement 
$\mathsf {TAC}+\mathsf {B}\Sigma ^0_2$ where 
$\mathsf {TAC}$ is the tree antichain theorem introduced by Conidis [6]. We show that 
${\mathsf {CAC\ for\ trees}}$ is computationally very weak, in that it admits probabilistic solutions.
