Stationary set
Encyclopedia
In mathematics
, particularly in set theory
and model theory
, there are at least three notions of stationary set:
is a cardinal
of uncountable cofinality
,
and 
intersects
every club set
in
then 
is called a stationary set. 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. Because if 
is any club set, then 
is a club set because the intersection of two club sets is club. 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.
, 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.
, 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 
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, particularly in set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
and model theory
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
, there are at least three notions of stationary set:
Classical notion
If is a cardinalCardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
of uncountable cofinality
Cofinality
In mathematics, especially in order theory, the cofinality cf of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A....
,
and intersectsIntersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
every club set
Club set
In mathematics, particularly in mathematical logic and set theory, a club set is a subset of a limit ordinal which is closed under the order topology, and is unbounded relative to the limit ordinal...
in
then is called a stationary set. 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 theoryThin set (Serre)
In mathematics, a thin set in the sense of Serre, named after Jean-Pierre Serre, is a certain kind of subset constructed in algebraic geometry over a given field K, by allowed operations that are in a definite sense 'unlikely'. The two fundamental ones are: solving a polynomial equation that may or...
.
If
is a stationary set and is a club set, then their intersection is also stationary. Because if is any club set, then is a club set because the intersection of two club sets is club. Thus is non empty. Therefore must be stationary.See also: Fodor's lemma
Fodor's lemma
In mathematics, particularly in set theory, Fodor's lemma states the following:If \kappa is a regular, uncountable cardinal, S is a stationary subset of \kappa, and f:S\rightarrow\kappa is regressive In mathematics, particularly in set theory, Fodor's lemma states the following:If \kappa is a...
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 regularRegular cardinal
In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. So, crudely speaking, a regular cardinal is one which cannot be broken into a smaller collection of smaller parts....
and
is stationary. Then can be partitioned into many disjoint stationary sets. This result is due to SolovayRobert M. Solovay
Robert Martin Solovay is an American mathematician specializing in set theory.Solovay earned his Ph.D. from the University of Chicago in 1964 under the direction of Saunders Mac Lane, with a dissertation on A Functorial Form of the Differentiable Riemann–Roch theorem...
. If
is a successor cardinalSuccessor cardinal
In the theory of cardinal numbers, we can define a successor operation similar to that in the ordinal numbers. This coincides with the ordinal successor operation for finite cardinals, but in the infinite case they diverge because every infinite ordinal and its successor have the same cardinality...
, this result is due to Ulam and is easily shown by means of what is called an Ulam matrix.
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 : . This notion is due to Thomas JechThomas Jech
Thomas J. Jech is a mathematician specializing in set theory who was at Penn State for more than 25 years. He was educated at Charles University and is now at the of the Academy of Sciences of the Czech Republic....
. 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
There is yet a third notion, model theoretic in nature and sometimes referred to as generalized stationarity. This notion is probably due to MagidorMenachem Magidor
Menachem Magidor is an Israeli mathematician who specializes in mathematical logic, in particular set theory. He served as President of the Hebrew University of Jerusalem.- Biography :Menachem Magidor was born in Petah Tikva on January 24, 1946....
, Foreman
Matthew Foreman
Matthew Dean Foreman is a set theorist at University of California, Irvine. He has made contributions in widely varying areas of set theory, including descriptive set theory, forcing, and infinitary combinatorics....
and Shelah
Saharon Shelah
Saharon Shelah is an Israeli mathematician. He is a professor of mathematics at the Hebrew University of Jerusalem and Rutgers University in New Jersey.-Biography:...
and has also been used prominently by Woodin
W. Hugh Woodin
William Hugh Woodin is an American mathematician and set theorist at University of California, Berkeley. He has made many notable contributions to the theory of inner models and determinacy. A type of large cardinal, the Woodin cardinal, bears his name.-Biography:Born in Tucson, Arizona, 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 .