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.

is a cardinal of uncountable cofinality,

intersects every club set in

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.

is a stationary set and

is a club set, then their intersection

See also: Fodor's lemma The restriction to uncountable cofinality is in order to avoid trivialities: Suppose

This is no longer the case if the cofinality of

many disjoint stationary sets.

This result is due to Solovay.

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

contains a closed subset of order type

There is also a notion of stationary subset of

This notion is due to Thomas Jech.

is stationary if and only if it meets every club, where a club subset of

and closed under union of chains of length at most

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

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.

is club (closed and unbounded) if and only if there is a function

is the collection of finite subsets of

if and only if it meets every club subset of

To see the connection with model theory, notice that if

must contain an elementary substructure of

is stationary if and only if for any such structure

there is an elementary substructure of