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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2408.17393 |
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| _version_ | 1866910584605245440 |
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| author | Schweber, Noah |
| author_facet | Schweber, Noah |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_17393 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Strong reducibilities and set theory Schweber, Noah Logic 03D45 (Primary) 03D60, 03E15 (Secondary) 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. |
| title | Strong reducibilities and set theory |
| topic | Logic 03D45 (Primary) 03D60, 03E15 (Secondary) |
| url | https://arxiv.org/abs/2408.17393 |