标题： 强可约性与集合论 显示英文标题

标题： Strong reducibilities and set theory

摘要： 我们研究集合论中的Medvedev可约性——具体来说，是强迫和大基数假说。 回答Hamkins和Li的问题\cite{HaLi}，我们表明可数序数的Medvedev度在多种方式下远非线性有序，我们在此的主要结果是存在一个序数的闭无界集，它在Medvedev可约性下是一个反链。 然后我们在适当集合论假定下，将这些结果推广到任意“合理定义”的可约性。 接着我们从序数转向一般结构。 我们表明上述一些结果可以用于刻画Vaught猜想的反例；另一个结果适用于所有情况，为任何合理的结构类分配一个序数和该类上的“测度”。 最后我们讨论了一些未来研究的方向。 显示英文摘要

摘要： 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.