Fodor's lemma

In mathematics, particularly in set theory, Fodor's lemma states the following:

If $$kappa is a regular, uncountable cardinal, $$S is a stationary subset of $$kappa, and $$f:kappa ightarrowkappa is regressive on $$S (that is, $$f(alpha) for any $$alphain S, $$alpha eq 0) then there is some $$gamma and some stationary $$S_0subseteq S such that $$f(alpha)=gamma for any $$alphain S_0. In modern parlance, the nonstationary ideal is "normal".

A proof of Fodor's lemma is as follows:

If we let $$f^{-1}:kappa ightarrow P(S) be the inverse of $$f restricted to $$S then Fodor's lemma is equivalent to the claim that for any function such that $$alphain f(kappa) ightarrow alpha>f(alpha) there is some $$alphain S such that $$f^{-1}(alpha) is stationary.

Then if Fodor's lemma is false, for every $$alphain S there is some club set $$C_alpha such that $$C_alphacap f^{-1}(alpha)=emptyset. Let $$C=Delta_{alpha. The club sets are closed under diagonal intersection, so $$C is also club and therefore there is some $$alphain Scap C. Then $$alphain C_eta for each $$eta, and so there can be no $$eta such that $$alphain f^{-1}(eta), so $$f(alpha)geqalpha, a contradiction.

The lemma was first proved by the Hungarian set theorist, Géza Fodor in 1952.

References

* 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 [http://www.math.rutgers.edu/~sthomas/book.ps]