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A Mitchell-like order for Ramsey and Ramsey-like cardinals

Volume 248 / 2020

Erin Carmody, Victoria Gitman, Miha E. Habič Fundamenta Mathematicae 248 (2020), 1-32 MSC: Primary 03E55. DOI: 10.4064/fm701-3-2019 Published online: 16 September 2019

Abstract

Smallish large cardinals $\kappa $ are often characterized by the existence of a collection of filters on $\kappa $, each of which is an ultrafilter on the subsets of $\kappa $ of some transitive $\rm ZFC ^-$-model of size $ \kappa $. We introduce a Mitchell-like order for Ramsey and Ramsey-like cardinals, ordering such collections of small filters. We show that the Mitchell-like order and the resulting notion of rank have all the desirable properties of the Mitchell order on normal measures on a measurable cardinal. The Mitchell-like order behaves robustly with respect to forcing constructions. We show that extensions with the cover and approximation properties cannot increase the rank of a Ramsey or Ramsey-like cardinal. We use the results about extensions with the cover and approximation properties together with recently developed techniques about soft killing of large-cardinal degrees by forcing to softly kill the ranks of Ramsey and Ramsey-like cardinals.

Authors

  • Erin CarmodyMathematics Department
    Fordham University
    Bronx, NY 10458, U.S.A.
    e-mail
  • Victoria GitmanMathematics Program
    CUNY Graduate Center
    The City University of New York 365 Fifth Avenue
    New York, NY 10016, U.S.A.
    http:victoriagitman.github.io/
    e-mail
  • Miha E. HabičDepartment of Logic
    Faculty of Arts
    Charles University
    Celetná 20
    116 42 Praha 1, Czech Republic
    http:mhabic.github.io/
    e-mail

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