Karagila, A. orcid.org/0000-0003-1289-0904 and Yuan, J. (2026) Critical embeddings. Monatshefte für Mathematik. ISSN: 0026-9255
Abstract
Hayut and the first author isolated the notion of a critical cardinal in [1]. In this work, we answer several questions raised in the original paper. We show that it is consistent for a critical cardinal not to have any ultrapower elementary embeddings, as well as that it is consistent that no target model is closed. We also prove that if κ is a critical point by any ultrapower embedding, then it is the critical point of an ultrapower embedding by a normal measure. The paper concludes by presenting several open questions of interest in the study of critical cardinals.
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| Item Type: | Article |
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| Authors/Creators: |
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| Copyright, Publisher and Additional Information: | © The Author(s) 2026. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. |
| Keywords: | Critical cardinals; Elementary embeddings; Axiom of choice; Symmetric extensions |
| Dates: |
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| Institution: | The University of Leeds |
| Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) |
| Funding Information: | Funder Grant number Medical Research Council (MRC) **KRISTAL** MR/T021705/2 |
| Date Deposited: | 07 Aug 2026 10:03 |
| Last Modified: | 19 Oct 2026 15:28 |
| Status: | Published online |
| Publisher: | Springer Nature |
| Identification Number: | 10.1007/s00605-026-02215-0 |
| Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:244229 |
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