{"title":"Clarifying ordinals","authors":"Noah Schweber","doi":"arxiv-2408.10367","DOIUrl":null,"url":null,"abstract":"We use forcing over admissible sets to show that, for every ordinal $\\alpha$\nin a club $C\\subset\\omega_1$, there are copies of $\\alpha$ such that the\nisomorphism between them is not computable in the join of the complete\n$\\Pi^1_1$ set relative to each copy separately. Assuming $\\mathsf{V=L}$, this\nis close to optimal; on the other hand, assuming large cardinals the same (and\nmore) holds for every projective functional.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.10367","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We use forcing over admissible sets to show that, for every ordinal $\alpha$
in a club $C\subset\omega_1$, there are copies of $\alpha$ such that the
isomorphism between them is not computable in the join of the complete
$\Pi^1_1$ set relative to each copy separately. Assuming $\mathsf{V=L}$, this
is close to optimal; on the other hand, assuming large cardinals the same (and
more) holds for every projective functional.
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