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Critical cardinals

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Abstract

We introduce the notion of a critical cardinal as the critical point of sufficiently strong elementary embedding between transitive sets. Assuming the Axiom of Choice this is equivalent to measurability, but it is wellknown that Choice is necessary for the equivalence. Oddly enough, this central notion was never investigated on its own before. We prove a technical criterion for lifting elementary embeddings to symmetric extensions, and we use this to show that it is consistent relative to a supercompact cardinal that there is a critical cardinal whose successor is singular.

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Author information

Author notes

  1. Yair Hayut

    Present address: Current address: Kurt Gödel Research Center for Mathematical Logic, University of Vienna, Währinger Strasse 25, 1090, Vienna, Austria

Authors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 69978, Israel

    Yair Hayut

  2. School of Mathematics, University of East Anglia Norwich, Norwich, NR4 7TJ, United Kingdom

    Asaf Karagila

Authors
  1. Yair Hayut
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  2. Asaf Karagila
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Corresponding author

Correspondence to Asaf Karagila.

Additional information

The second author was partially supported by the Austrian Science Foundation FWF, grant I 3081-N35, and by the Royal Society Newton International Fellowship, grant no. NF170989.

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Hayut, Y., Karagila, A. Critical cardinals. Isr. J. Math. 236, 449–472 (2020). https://doi.org/10.1007/s11856-020-1998-8

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  • DOI: https://doi.org/10.1007/s11856-020-1998-8

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