A club guessing toolbox I
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2022
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| _version_ | 1866909467281457152 |
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| author | Inamdar, Tanmay Rinot, Assaf |
| author_facet | Inamdar, Tanmay Rinot, Assaf |
| contents | Club guessing principles were introduced by Shelah as a weakening of Jensen's diamond. Most spectacularly, they were used to prove Shelah's ZFC bound on the power of the first singular cardinal.
These principles have found many other applications: in cardinal arithmetic and PCF theory; in the construction of combinatorial objects on uncountable cardinals such as Jonsson algebras, strong colourings, Souslin trees, and pathological graphs; to the non-existence of universals in model theory; to the non-existence of forcing axioms at higher uncountable cardinals; and many more.
In this paper, the first part of a series, we survey various forms of club-guessing that have appeared in the literature, and then systematically study the various ways in which a club-guessing sequences can be improved, especially in the way the frequency of guessing is calibrated.
We include an expository section intended for those unfamiliar with club-guessing and which can be read independently of the rest of the article. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_03969 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A club guessing toolbox I Inamdar, Tanmay Rinot, Assaf Logic Club guessing principles were introduced by Shelah as a weakening of Jensen's diamond. Most spectacularly, they were used to prove Shelah's ZFC bound on the power of the first singular cardinal. These principles have found many other applications: in cardinal arithmetic and PCF theory; in the construction of combinatorial objects on uncountable cardinals such as Jonsson algebras, strong colourings, Souslin trees, and pathological graphs; to the non-existence of universals in model theory; to the non-existence of forcing axioms at higher uncountable cardinals; and many more. In this paper, the first part of a series, we survey various forms of club-guessing that have appeared in the literature, and then systematically study the various ways in which a club-guessing sequences can be improved, especially in the way the frequency of guessing is calibrated. We include an expository section intended for those unfamiliar with club-guessing and which can be read independently of the rest of the article. |
| title | A club guessing toolbox I |
| topic | Logic |
| url | https://arxiv.org/abs/2207.03969 |