Strong reducibilities and set theory

TL;DR

This paper explores the complexity of Medvedev reducibility within set theory, revealing that the degrees of countable ordinals form complex, non-linear structures and extending these findings to broader classes of structures under set-theoretic assumptions.

Contribution

It demonstrates that Medvedev degrees of countable ordinals are highly non-linear, providing new insights into their structure and extending results to general structures and reducibilities.

Findings

Medvedev degrees of countable ordinals are not linearly ordered.

There exists a club of ordinals forming an antichain under Medvedev reducibility.

Results apply to characterizations of counterexamples to Vaught's conjecture.

Abstract

We study Medvedev reducibility in the context of set theory -- specifically, forcing and large cardinal hypotheses. Answering a question of Hamkins and Li \cite{HaLi}, we show that the Medvedev degrees of countable ordinals are far from linearly ordered in multiple ways, our main result here being that there is a club of ordinals which is an antichain with respect to Medvedev reducibility. We then generalize these results to arbitrary ``reasonably-definable" reducibilities, under appropriate set-theoretic hypotheses. We then turn from ordinals to general structures. We show that some of the results above yield characterizations of counterexamples to Vaught's conjecture; another applies to all situations, assigning an ordinal to any reasonable class of structures and ``measure" on that class. We end by discussing some directions for future research.