Stationary set

In mathematics, specifically set theory and model theory, a stationary set is a set 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 measure 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 notionEdit

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  .

Jech's notionEdit

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 notionEdit

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  .

ReferencesEdit

  1. ^ Jech (2003) p.91
  • Foreman, Matthew (2002) Stationary sets, Chang's Conjecture and partition theory, in Set Theory (The Hajnal Conference) DIMACS Ser. Discrete Math. Theoret. Comp. Sci., 58, Amer. Math. Soc., Providence, RI. pp. 73–94. File at [1]
  • Friedman, Harvey (1974). "On closed sets of ordinals". Proc. Am. Math. Soc. 43: 190–192. doi:10.2307/2039353. Zbl 0299.04003.
  • Jech, Thomas (2003). Set Theory. Springer Monographs in Mathematics (Third Millennium ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-44085-7. Zbl 1007.03002.

External linksEdit