Stationary set explained
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 notion
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
(S\capC)\capD=S\cap(C\capD)
is nonempty. 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 moreover
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 notion
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
:
[X]λ=\{Y\subseteqX:|Y|=λ\}
. 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
| \omega |
S\subseteq[\omega | |
| 1] |
is stationary if and only if
is stationary in
.
The appropriate version of Fodor's lemma also holds for this notion.
Generalized 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.
Now let
be a nonempty set. A set
is club (closed and unbounded) if and only if there is a function
such that
C=\{z:F[[z]<\omega]\subseteqz\}
. 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
- 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 https://web.archive.org/web/20050515042933/http://math.uci.edu/sub2/Foreman/homepage/hajfin.ps
- Friedman . Harvey . Harvey Friedman . On closed sets of ordinals . 0299.04003 . Proc. Am. Math. Soc. . 43 . 190–192 . 1974 . 1 . 10.2307/2039353 . 2039353 . free .
- Book: Jech . Thomas . Thomas Jech . Set Theory . Third Millennium . . Berlin, New York . Springer Monographs in Mathematics . 978-3-540-44085-7 . 2003 . 1007.03002 .
Notes and References
- Jech (2003) p.91