In mathematics, particularly in set theory, Fodor's lemma states the following: Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...

If κ is a regular, uncountable cardinal, S is a stationary subset of κ, and is regressive on S (that is, f(α) < α for any ) then there is some γ and some stationary such that f(α) = γ for any . In mathematics, an uncountable set is a set which is not countable. ... In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ... In mathematics, particularly in set theory, if is a cardinal, , and intersects every club in , then is called a stationary set. ...

A proof of Fodor's lemma is as follows:

If we let be the inverse of f restricted to S then Fodor's lemma is equivalent to the claim that for any function such that there is some such that f ^{− 1}(α) is stationary. Inverse typically means the opposite of something. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...

Then if Fodor's lemma is false, for every there is some club set C_α such that . Let C = Δ_{α < κ}C_α. The club sets are closed under diagonal intersection, so C is also club and therefore there is some . Then for each β < α, and so there can be no β < α such that , so , a contradiction. A club set is a subset of a limit ordinal which is closed under the order topology, and is unbounded. ... Broadly speaking, a contradiction is when two or more statements, ideas, or actions are seen as incompatible. ...

References

• 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] (http://www.math.rutgers.edu/~sthomas/book.ps)

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