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  \kappa \, is a cardinal of uncountable cofinality,  S \subseteq \kappa \,, and  S \, intersects every club set in  \kappa \,, then  S \, 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  S \, is a stationary set and  C \, is a club set, then their intersection  S \cap C \, is also stationary. This is because if  D \, is any club set, then  C \cap D \, is a club set, thus  (S \cap C) \cap D = S \cap (C  \cap D) \, is non empty. Therefore  (S \cap C) \, must be stationary.

See also: Fodor's lemma

The restriction to uncountable cofinality is in order to avoid trivialities: Suppose \kappa has countable cofinality. Then S\subset\kappa is stationary in \kappa if and only if \kappa\setminus S is bounded in \kappa. In particular, if the cofinality of \kappa is \omega=\aleph_0, then any two stationary subsets of \kappa have stationary intersection.

This is no longer the case if the cofinality of \kappa is uncountable. In fact, suppose \kappa is regular and S\subset\kappa is stationary. Then S can be partitioned into \kappa many disjoint stationary sets. This result is due to Solovay. If \kappa 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 \beta, every stationary subset of \omega_1 contains a closed subset of order type \beta (Friedman).

Jech's notion[edit]

There is also a notion of stationary subset of [X]^\lambda, for \lambda a cardinal and X a set such that |X|\ge\lambda, where [X]^\lambda is the set of subsets of X of cardinality \lambda: [X]^\lambda=\{Y\subset X:|Y|=\lambda\}. This notion is due to Thomas Jech. As before, S\subset[X]^\lambda is stationary if and only if it meets every club, where a club subset of [X]^\lambda is a set unbounded under \subset and closed under union of chains of length at most \lambda. These notions are in general different, although for X=\omega_1 and \lambda=\aleph_0 they coincide in the sense that S\subset[\omega_1]^\omega is stationary if and only if S\cap\omega_1 is stationary in \omega_1.

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 X be a nonempty set. A set C\subset{\mathcal P}(X) is club (closed and unbounded) if and only if there is a function F:[X]^{<\omega}\to X such that C=\{z:F[[z]^{<\omega}]\subset z\}. Here, [y]^{<\omega} is the collection of finite subsets of y.

S\subset{\mathcal P}(X) is stationary in {\mathcal P}(X) if and only if it meets every club subset of {\mathcal P}(X).

To see the connection with model theory, notice that if M is a structure with universe X in a countable language and F is a Skolem function for M, then a stationary S must contain an elementary substructure of M. In fact, S\subset{\mathcal P}(X) is stationary if and only if for any such structure M there is an elementary substructure of M that belongs to S.

References[edit]

  1. ^ Jech (2003) p.91

External links[edit]