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  • CLOSURE PROPERTIES OF MEASURABLE ULTRAPOWERS

  • Part of
  • PHILIPP LÜCKE, SANDRA MÜLLER
  • Journal:
    The Journal of Symbolic Logic / Volume 86 / Issue 2 / June 2021
    Published online by Cambridge University Press:
    06 May 2021, pp. 762-784
    Print publication:
    June 2021
    • Article
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  • We study closure properties of measurable ultrapowers with respect to Hamkin’s notion of freshness and show that the extent of these properties highly depends on the combinatorial properties of the underlying model of set theory. In one direction, a result of Sakai shows that, by collapsing a strongly compact cardinal to become the double successor of a measurable cardinal, it is possible to obtain a model of set theory in which such ultrapowers possess the strongest possible closure properties. In the other direction, we use various square principles to show that measurable ultrapowers of canonical inner models only possess the minimal amount of closure properties. In addition, the techniques developed in the proofs of these results also allow us to derive statements about the consistency strength of the existence of measurable ultrapowers with non-minimal closure properties.