If is a regular, uncountable cardinal, is a stationary subset of , and is regressive (that is, for any , ) then there is some and some stationary such that for any . In modern parlance, the nonstationary ideal is normal.
We can assume that (by removing 0, if necessary). If Fodor's lemma is false, for every there is some club set such that . Let . The club sets are closed under diagonal intersection, so is also club and therefore there is some . Then for each , and so there can be no such that , so , a contradiction.
The lemma was first proved by the Hungarian set theorist, Géza Fodor in 1956. It is sometimes also called "The Pressing Down Lemma".
- G. Fodor, Eine Bemerkung zur Theorie der regressiven Funktionen, Acta Sci. Math. Szeged, 17(1956), 139-142.
- Karel Hrbacek & Thomas Jech, Introduction to Set Theory, 3rd edition, Chapter 11, Section 3.
- Mark Howard, Applications of Fodor's Lemma to Vaught's Conjecture. Ann. Pure and Appl. Logic 42(1): 1-19 (1989).
- Simon Thomas, The Automorphism Tower Problem. PostScript file at