Weihrauch reducibility between Ramsey-type theorems and well-ordering principles at the level of $Σ^0_2$-induction: A pilot study
Lorenzo Carlucci, Giordano Celli
TL;DR
This work analyzes Weihrauch reductions between the well-ordering preservation principle for the operator $X \mapsto X^\omega$ and the Ordered Ramsey Theorem. The authors prove a precise equivalence: $\mathsf{ORT}_{\mathbb N} \equiv_W \mathsf{ECT} \times \mathsf{WOP}(X\mapsto X^\omega)$, by establishing a matching lower bound via a colorings-based encoding and an upper bound via a parameterized reduction that uses lexicographic variants. They further show that $\mathsf{WOP}(X\mapsto X^\omega)$ is Weihrauch-incomparable with $(\mathsf{LPO}')^*$ and that $\mathsf{ECT}$ is strictly weaker than $\mathsf{WOP}(X\mapsto X^\omega)$, while $\mathsf{WOP}(X\mapsto X^\omega)$ is strictly weaker than $\mathsf{ECT} \times \mathsf{WOP}(X\mapsto X^\omega)$. This completes a refined splitting of $\mathsf{ORT}_{\mathbb N}$ and clarifies the Weihrauch degree structure around $\Sigma^0_2$-induction, providing a first example of reducing a Ramsey-type principle to a WOP and suggesting further exploration of WOPs in the Weihrauch framework.
Abstract
We study the relations under Weihrauch reducibility of the well-ordering preservation principle for the operator $X \mapsto X^ω$ and the Ordered Ramsey Theorem. Both principles are known to be equivalent to $Σ^0_2$-induction in Reverse Mathematics. We show that the Ordered Ramsey Theorem is Weihrauch-equivalent to the parallel product of the well-ordering preservation principle for the operator $X \mapsto X^ω$ and the Eventually Constant Tail principle. By previous work from Pauly, Pradic and Soldà, the Ordered Ramsey Theorem is known to be Weihrauch-equivalent to the parallel product of the Eventually Constant Tail principle and the parallelization of the jump of the Limited Principle of Omniscience. We show that the latter pinciple and the well-ordering preservation principle for $X \mapsto X^ω$ are Weihrauch-incomparable.