Strong reducibilities and set theory

Abstract

We study Medvedev reducibility in the context of set theory — specifically, forcing and large cardinal hypotheses. Answering a question of Hamkins and Li [6], 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.