Mantel test: Difference between revisions

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In [[mathematics]], particularly in [[set theory]], '''Fodor's lemma''' states the following:
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If <math>\kappa</math> is a [[Regular cardinal|regular]], [[uncountable]] [[Cardinal number|cardinal]], <math>S</math> is a [[stationary set|stationary subset]] of <math>\kappa</math>, and <math>f:S\rightarrow\kappa</math> is regressive (that is, <math>f(\alpha)<\alpha</math> for any <math>\alpha\in S</math>, <math>\alpha\neq 0</math>) then there is some <math>\gamma</math> and some stationary <math>S_0\subseteq S</math> such that <math>f(\alpha)=\gamma</math> for any <math>\alpha\in S_0</math>. In modern parlance, the nonstationary ideal is ''normal''.
 
==Proof==
We can assume that <math>0\notin S</math> (by removing 0, if necessary).
If Fodor's lemma is false, for every <math>\alpha<\kappa</math> there is some [[club set]] <math>C_\alpha</math> such that <math>C_\alpha\cap f^{-1}(\alpha)=\emptyset</math>. Let <math>C=\Delta_{\alpha<\kappa} C_\alpha</math>. The club sets are closed under [[diagonal intersection]], so <math>C</math> is also club and therefore there is some <math>\alpha\in S\cap C</math>. Then <math>\alpha\in C_\beta</math> for each <math>\beta<\alpha</math>, and so there can be no <math>\beta<\alpha</math> such that <math>\alpha\in f^{-1}(\beta)</math>, so <math>f(\alpha)\geq\alpha</math>, a [[contradiction]].
 
The lemma was first proved by the Hungarian set theorist, [[Géza Fodor (mathematician)|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 [[Stationary_set#Generalized_notion|general notion]] of stationary set.
 
== References ==
* G. Fodor, Eine Bemerkung zur Theorie der regressiven Funktionen, [[Acta Scientiarum Mathematicarum|''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 [http://www.math.rutgers.edu/~sthomas/book.ps]
 
{{PlanetMath attribution|id=3232|title=Fodor's lemma}}
 
[[Category:Set theory]]
[[Category:Lemmas]]
[[Category:Articles containing proofs]]

Latest revision as of 20:51, 24 June 2014

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