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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bilinsky Eilon, Gitik Moti: A model with a measurable which does not carry a normal measure. Arch. Math. Logic, 51, 863–876 (2012)
Anne Fernengel and Peter Koepke, An Easton-like theorem for Zermelo–Fraenkel set theory without choice, ArXiv e-prints, 1607.00205 (2016), pp. 57.
Gitik M.: All uncountable cardinals can be singular. Israel J. Math., 35, 61–88 (1980)
Serge Grigorieff, Intermediate submodels and generic extensions in set theory, Ann. Math. (2), 101 (1975), 447–490.
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).
Karagila Asaf: Embedding orders into the cardinals with \({\mathsf{DC}_\kappa}\). Fund. Math., 226, 143–156 (2014)
Asaf Karagila, Iterating symmetric extensions, ArXiv e-prints, 1606.06718 (2016), pp. 32. , submitted.
Author information
Authors and Affiliations
Corresponding author
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.
Rights and permissions
About this article
Cite this article
Karagila, A. Fodor’s lemma can fail everywhere. Acta Math. Hungar. 154, 231–242 (2018). https://doi.org/10.1007/s10474-017-0768-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-017-0768-5
Key words and phrases
- symmetric extension
- Fodor’s lemma
- closed and unbounded filter
- iterated symmetric extension
- the axiom of choice