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Fodor's lemma
Concept in mathematical set theory From Wikipedia, the free encyclopedia
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In mathematics, particularly in set theory, Fodor's lemma states the following:
![]() | This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. (January 2020) |
If is a regular, uncountable cardinal, is a stationary subset of , and is regressive (that is, for any , ) then there is some and some stationary such that for any . In modern parlance, the nonstationary ideal is normal.
The lemma was first proved by the Hungarian set theorist, Géza Fodor in 1956. It is sometimes also called "The Pressing Down Lemma".
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Proof
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We can assume that (by removing 0, if necessary). If Fodor's lemma is false, for every there is some club set such that . Let . The club sets are closed under diagonal intersection, so is also club and therefore there is some . Then for each , and so there can be no such that , so , a contradiction.
Fodor's lemma also holds for Thomas Jech's notion of stationary sets as well as for the general notion of stationary set.
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Fodor's lemma for trees
Another related statement, also known as Fodor's lemma (or Pressing-Down-lemma), is the following:
For every non-special tree and regressive mapping (that is, , with respect to the order on , for every , ), there is a non-special subtree on which is constant.
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References
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