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See also

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	In ~~[[mathematics]]~~, ~~particularly in [[set theory]]~~, ~~'''Fodor's lemma''' states~~ the ~~following:~~		Boost your online visibility: In this regard, you can create, promote and boost your website visibility to get more success and profit as well. One of the largest advantages that Wordpress has over Blogger is the fact that you can totally customize any blog using plug-ins. You'll spend maybe $30-40 on a prepaid card from Walgreens, empty it paying for trials, and get back $100-200 in offer rewards, a worthy investment. Doing so perpetuates you as a brand, and you can integrate your other social profiles in there. Among the list of worlds most significant and most preferred Internet hosting enterprise is the HostGator. To manage their domains with [http://www.google.com/search?q=efficiency&btnI=lucky efficiency] from one centre and be able to access on the control panels they would do anything to get these coupons. And when it comes to this, there's nothing better than web hosting reviews to keep yourself informed - but of course, you'll need to know how to make proper use of them in order to get the full benefits that they offer. They have the most professional and complete webhosting services and plans to pick from. If you are looking for web hosting packages, you'll be surprised to find a great number of web hosting companies and even individuals offering this type of services. Kontera is text link ads, these ads are in the text underlined.<br><br><br><br>Irregardless of the complexity of my query, I was assisted.HostGator has a quite straightforward-to-use interfaceWhen HostGator passed my purchaser provider checks, I checked out their Cpanel (the person interface) to see if it was as uncomplicated-to-use as Bluehost. The main reasons I use HostGator for my webhosting is ease of use, amazing support, and good prices. Webhosts are companies that supply space on a server they own or lease for use by their clients as well as providing internet connectivity. When I first started blogging I decided to use Blogger as it doesn't require you to self host so I needed no upfront investment to test the waters. All of them promise to offer you an easier time when putting up your personal website. Some of these items have a cost involved. You get some kind of discount when signing up with them, which is really fantastic. Inc.5000 comprises of listings of the [http://www.bing.com/search?q=fastest+growing&form=MSNNWS&mkt=en-us&pq=fastest+growing fastest growing] companies In the United States Of America When you have just about any issues concerning where along with tips on how to employ [http://orichinese.com/groups/rv-site-builder-hostgator/ Hostgator Promo Code], you'll be able to call us with the web page. .

	~~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]~~]

en>Gaurav: Changed {{cite article}} to {{cite arXiv}}, fixed text

2014-01-30T04:27:01Z

Changed {{cite article}} to {{cite arXiv}}, fixed text

New page

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

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]]

Mantel test - Revision history

213.33.9.20: /* See also */

en>Gaurav: Changed {{cite article}} to {{cite arXiv}}, fixed text