Fodor's lemma

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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:S\rightarrow\kappa is regressive (that is, f(\alpha)<\alpha for any \alpha\in S, \alpha\neq 0) then there is some \gamma and some stationary S_0\subseteq S such that f(\alpha)=\gamma for any \alpha\in S_0. In modern parlance, the nonstationary ideal is normal.

Proof[edit]

We can assume that 0\notin S (by removing 0, if necessary). If Fodor's lemma is false, for every \alpha<\kappa there is some club set C_\alpha such that C_\alpha\cap f^{-1}(\alpha)=\emptyset. Let C=\Delta_{\alpha<\kappa} C_\alpha. The club sets are closed under diagonal intersection, so C is also club and therefore there is some \alpha\in S\cap C. Then \alpha\in C_\beta for each \beta<\alpha, and so there can be no \beta<\alpha such that \alpha\in f^{-1}(\beta), so f(\alpha)\geq\alpha, 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 also holds for Thomas Jech's notion of stationary sets as well as for the general notion of stationary set.

References[edit]

  • 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 [1]

This article incorporates material from Fodor's lemma on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.