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".
Fodor's lemma for trees
Another related statement, also known as Fodor's lemma (or Pressing-Down-lemma), is the following:
For every non-special tree and regressive mapping (that is, , with respect to the order on , for every ), there is a non-special subtree on which is constant.
- 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 
- S. Todorcevic, Combinatorial dichotomies in set theory. pdf at