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,

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

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]