Fodor's theorem

The set of Fodor (also: Pressing down lemma ) is a set of the set theory , the 1956 by Hungarian mathematician the Géza Fodor was discovered. It says that for certain functions there are always large (i.e. stationary ) subsets on which they only take on one value.

statement

Let be a stationary subset of a regular , uncountable cardinal number . Is a regressive function, i. H. holds for all , then there is a stationary set on which is constant, i.e. H. there is one such that applies to all . ${\displaystyle S\subseteq \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle f\colon S\to \kappa }$ ${\displaystyle f(\alpha )<\alpha }$ ${\displaystyle \alpha \in S\setminus \{0\}}$ ${\displaystyle T\subseteq S}$ ${\displaystyle f}$ ${\displaystyle \gamma \in \kappa }$ ${\displaystyle f(\alpha )=\gamma }$ ${\displaystyle \alpha \in T}$

proof

Assumption that the statement does not apply: Then the set would be non-stationary for each . Therefore the complements are respectively supersets of club sets , i.e. elements of the club filter . This is complete compared to diagonal cuts , so the following applies . Since is stationary, is . But for :, so for everyone . This is in contradiction to regressivity. So the assumption is wrong that there is such a stationary set. ${\displaystyle \gamma \in \kappa }$ ${\displaystyle D_{\gamma }=\{\alpha \in \kappa \mid f(\alpha )=\gamma \}}$ ${\displaystyle D_{\gamma }^{\mathrm {C} }}$ ${\displaystyle {\mathcal {C}}_{\kappa }}$ ${\displaystyle \textstyle C=\bigtriangleup _{\gamma <\kappa }D_{\gamma }^{\mathrm {C} }=\{\alpha \in \kappa \mid \alpha \in \bigcap _{\gamma <\alpha }D_{\gamma }^{\mathrm {C} }\}\in {\mathcal {C}}_{\kappa }}$ ${\displaystyle S}$ ${\displaystyle S\cap C\neq \emptyset }$ ${\displaystyle \alpha \in S\cap C}$ ${\displaystyle \forall \gamma <\alpha :\alpha \notin D_{\gamma }}$ ${\displaystyle f(\alpha )\neq \gamma }$ ${\displaystyle \gamma <\alpha }$

literature

• Fodor, Géza: A note on the theory of regressive functions , Acta Sci. Math. Szeged, 17 (1956), pp. 139-142.
• Jech, Thomas: Set Theory , Springer-Verlag Berlin Heidelberg (2006), ISBN 3-540-44085-2 .