Stationary set

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In mathematical set theory and model theory, a stationary set is one that is not too small in the sense that it intersects all club sets, and is analogous to a set of non-zero measure in set theory. There are at least three closely related notions of stationary set, depending on whether one is looking at subsets of an ordinal, or subsets of something of given cardinality, or a powerset.

Classical notion[edit]

If is a cardinal of uncountable cofinality, and intersects every club set in then is called a stationary set.[1] If a set is not stationary, then it is called a thin set. This notion should not be confused with the notion of a thin set in number theory.

If is a stationary set and is a club set, then their intersection is also stationary. This is because if is any club set, then is a club set, thus is non empty. Therefore, must be stationary.

See also: Fodor's lemma

The restriction to uncountable cofinality is in order to avoid trivialities: Suppose has countable cofinality. Then is stationary in if and only if is bounded in . In particular, if the cofinality of is , then any two stationary subsets of have stationary intersection.

This is no longer the case if the cofinality of is uncountable. In fact, suppose is regular and is stationary. Then can be partitioned into many disjoint stationary sets. This result is due to Solovay. If is a successor cardinal, this result is due to Ulam and is easily shown by means of what is called an Ulam matrix.

H. Friedman has shown that for every countable successor ordinal , every stationary subset of contains a closed subset of order type (Friedman).

Jech's notion[edit]

There is also a notion of stationary subset of , for a cardinal and a set such that , where is the set of subsets of of cardinality : . This notion is due to Thomas Jech. As before, is stationary if and only if it meets every club, where a club subset of is a set unbounded under and closed under union of chains of length at most . These notions are in general different, although for and they coincide in the sense that is stationary if and only if is stationary in .

The appropriate version of Fodor's lemma also holds for this notion.

Generalized notion[edit]

There is yet a third notion, model theoretic in nature and sometimes referred to as generalized stationarity. This notion is probably due to Magidor, Foreman and Shelah and has also been used prominently by Woodin.

Now let be a nonempty set. A set is club (closed and unbounded) if and only if there is a function such that . Here, is the collection of finite subsets of .

is stationary in if and only if it meets every club subset of .

To see the connection with model theory, notice that if is a structure with universe in a countable language and is a Skolem function for , then a stationary must contain an elementary substructure of . In fact, is stationary if and only if for any such structure there is an elementary substructure of that belongs to .

References[edit]

  1. ^ Jech (2003) p.91

External links[edit]