Acta Mathematica Hungarica

, Volume 154, Issue 1, pp 231–242 | Cite as

Fodor’s lemma can fail everywhere

Article
  • 38 Downloads

Abstract

We show that it is equiconsistent with \({\mathsf{ZF}}\) that Fodor’s lemma fails everywhere, and furthermore that the club filter on every regular cardinal is not even \({\sigma}\)-complete. Moreover, these failures can be controlled in a very precise manner.

Key words and phrases

symmetric extension Fodor’s lemma closed and unbounded filter iterated symmetric extension the axiom of choice 

Mathematics Subject Classification

primary 03E40 secondary 03E05 03E25 03E35 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bilinsky Eilon, Gitik Moti: A model with a measurable which does not carry a normal measure. Arch. Math. Logic, 51, 863–876 (2012)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Anne Fernengel and Peter Koepke, An Easton-like theorem for Zermelo–Fraenkel set theory without choice, ArXiv e-prints, 1607.00205 (2016), pp. 57.
  3. 3.
    Gitik M.: All uncountable cardinals can be singular. Israel J. Math., 35, 61–88 (1980)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Serge Grigorieff, Intermediate submodels and generic extensions in set theory, Ann. Math. (2), 101 (1975), 447–490.Google Scholar
  5. 5.
    Thomas J. Jech, The Axiom of Choice, Studies in Logic and the Foundations of Mathematics, Vol. 75, North-Holland Publishing Co. (Amsterdam–London); American Elsevier Publishing Co., Inc. (New York, 1973).Google Scholar
  6. 6.
    Karagila Asaf: Embedding orders into the cardinals with \({\mathsf{DC}_\kappa}\). Fund. Math., 226, 143–156 (2014)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Asaf Karagila, Iterating symmetric extensions, ArXiv e-prints, 1606.06718 (2016), pp. 32. , submitted.

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2017

Authors and Affiliations

  1. 1.Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat RamThe Hebrew University of JerusalemJerusalemIsrael
  2. 2.DMG/AlgebraTU WienWienAustria

Personalised recommendations