PCF theory is the name of a
mathematicalMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
theory, invented by
Saharon ShelahSaharon Shelah is an Israeli mathematician.-Life:Shelah received his Ph.D. in 1969 from the Hebrew University...
, that deals with the
cofinalityIn 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....
of the
ultraproductThe ultraproduct is a mathematical construction that appears mainly in abstract algebra and in model theory, a branch of mathematical logic. An ultraproduct is a quotient of the direct product of a family of structures. All factors need to have the same signature...
s of
ordered setAn ordered set or ordered collection may be any of the following.In computer science:* List * ArrayIn order theory in mathematics:...
s. It gives strong upper bounds on the cardinalities of
power setIn mathematics, given a set S, the power set of S, written , P, ℘ or 2
S, is the set of all subsets of S. In axiomatic set theory In mathematics, given a set S, the power set (or powerset) of S, written , P(S), ℘(S) or 2
S, is the set of all subsets of S. In...
s of
singularIn 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....
cardinalIn 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...
s. The abbreviation "PCF" stands for "possible
cofinalitiesIn 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....
".
If
A is an infinite set of
regular cardinalIn 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....
s,
D is an
ultrafilterIn the mathematical field of set theory, an ultrafilter on a set X is a collection of subsets of X that is a filter, that cannot be enlarged . An ultrafilter may be considered as a finitely additive measure. Then every subset of X is either considered "almost everything" or "almost nothing"...
on
A, then
we let denote the cofinality of the ordered set of functions
where the ordering defined as follows.
if .
pcf(
A) is the set of cofinalities that occur if we consider all ultrafilters on
A, that is,
Obviously, pcf(
A) consists of regular cardinals.
PCF theory is the name of a
mathematicalMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
theory, invented by
Saharon ShelahSaharon Shelah is an Israeli mathematician.-Life:Shelah received his Ph.D. in 1969 from the Hebrew University...
, that deals with the
cofinalityIn 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....
of the
ultraproductThe ultraproduct is a mathematical construction that appears mainly in abstract algebra and in model theory, a branch of mathematical logic. An ultraproduct is a quotient of the direct product of a family of structures. All factors need to have the same signature...
s of
ordered setAn ordered set or ordered collection may be any of the following.In computer science:* List * ArrayIn order theory in mathematics:...
s. It gives strong upper bounds on the cardinalities of
power setIn mathematics, given a set S, the power set of S, written , P, ℘ or 2
S, is the set of all subsets of S. In axiomatic set theory In mathematics, given a set S, the power set (or powerset) of S, written , P(S), ℘(S) or 2
S, is the set of all subsets of S. In...
s of
singularIn 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....
cardinalIn 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...
s. The abbreviation "PCF" stands for "possible
cofinalitiesIn 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....
".
Main definitions
If
A is an infinite set of
regular cardinalIn 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....
s,
D is an
ultrafilterIn the mathematical field of set theory, an ultrafilter on a set X is a collection of subsets of X that is a filter, that cannot be enlarged . An ultrafilter may be considered as a finitely additive measure. Then every subset of X is either considered "almost everything" or "almost nothing"...
on
A, then
we let denote the cofinality of the ordered set of functions
where the ordering defined as follows.
if .
pcf(
A) is the set of cofinalities that occur if we consider all ultrafilters on
A, that is,
Main results
Obviously, pcf(
A) consists of regular cardinals. Considering ultrafilters concentrated on elements of
A, we get that
. Shelah proved, that if , then pcf(
A) has a largest element, and there are subsets of
A such that for each ultrafilter
D on
A, is the least element θ of pcf(
A) such that . Consequently, .
Shelah also proved that if
A is an interval of regular cardinals (i.e.,
A is the set of all regular cardinals between two cardinals), then pcf(
A) is also an interval of regular cardinals and |pcf(
A)|<|
A|
+4.
This implies the famous inequality
assuming that ℵ
ωis
strong limitIn mathematics, limit cardinals are a type of cardinal number.With the cardinal successor operation defined, we can define a limit cardinal in analogy to that for limit ordinals: λ is a limit cardinal if and only if λ is neither a successor cardinal nor zero, i.e. we cannot "reach" λ by repeated...
.
If λ is an infinite cardinal, then
J<λ is the following ideal on
A.
B∈
J<λ if holds for every ultrafilter
D with
B∈
D. Then
J<λ is the ideal generated by the sets . There exist
scales, i.e., for every λ∈pcf(
A) there is a sequence of length λ of elements of which is both increasing and cofinal mod
J<λ. This implies that the cofinality of under pointwise dominance is max(pcf(
A)).
Another consequence is that if λ is singular and no regular cardinal less than λ is
JónssonIn set theory, a Jónsson cardinal is a certain kind of large cardinal number.An uncountable cardinal number κ is said to be Jónsson if for every function f: [κ]<ω → κ there is a set H of order type κ such that for each n, f restricted to n-element subsets of H omits at least one value...
, then there is a Jónsson algebra on λ
+. Specifically, there is a Jónsson algebra on ℵ
ω+1, which settles an old conjecture.
Unsolved problems
The most notorious conjecture in pcf theory states that |pcf(
A)|=|
A| holds for every set
A of regular cardinals with min(
A)
A). This would imply that if ℵω is strong limit, then the sharp bound
holds. The analogous bound
follows from Chang's conjecture (MagidorMenachem 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....
) or even from the nonexistence of a Kurepa treeIn set theory, a Kurepa tree is a tree of height , each of whose levels is at most countable, and has at least many branches. It was named after Yugoslav mathematician Đuro Kurepa. The existence of a Kurepa tree is independent of the axioms of ZFC. As Solovay showed, there are Kurepa trees in...
(ShelahSaharon Shelah is an Israeli mathematician.-Life:Shelah received his Ph.D. in 1969 from the Hebrew University...
).
A weaker, still unsolved conjecture states that if |A|A), then pcf(A) has no inaccessible limit point. This is equivalent to the statement that pcf(pcf(A))=pcf(A).
External links