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Peer-reviewedCC BY 4.0Monatshefte Fur Mathematik

Critical embeddings

  • Asaf Karagila1
  • Jiachen Yuan1
  1. 1School of Mathematics, University of Leeds, Leeds, LS2 9JT UK
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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 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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Cite

BibTeXRIS
BibTeX
@article{karagila2026critical,
  title = {{Critical embeddings}},
  author = {Asaf Karagila and Jiachen Yuan},
  journal = {Monatshefte Fur Mathematik},
  year = {2026},
  month = sep,
  doi = {10.1007/s00605-026-02215-0},
  url = {https://doi.org/10.1007/s00605-026-02215-0},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Critical embeddings
AU  - Asaf Karagila
AU  - Jiachen Yuan
PY  - 2026
DA  - 2026/09/18
JO  - Monatshefte Fur Mathematik
DO  - 10.1007/s00605-026-02215-0
UR  - https://doi.org/10.1007/s00605-026-02215-0
AB  - 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 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.
SN  - 0026-9255
SN  - 1436-5081
ER  -