Fodor’s lemma can fail everywhere


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.

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Correspondence to A. Karagila.

Additional information

This paper is part of the author’s Ph.D. thesis written at the Hebrew University of Jerusalem under the supervision of Prof. Menachem Magidor.

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Karagila, A. Fodor’s lemma can fail everywhere. Acta Math. Hungar. 154, 231–242 (2018).

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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