PCF theory

PCF theory

PCF theory is the name of a mathematical theory, introduced by Saharon Shelah (1978), that deals with the cofinality of the ultraproducts of ordered sets. It gives strong upper bounds on the cardinalities of power sets of singular cardinals, and has many more applications as well. The abbreviation "PCF" stands for "possible cofinalities".


Main definitions

If A is an infinite set of regular cardinals, D is an ultrafilter on A, then we let cf(\prod A/D) denote the cofinality of the ordered set of functions \prod A where the ordering is defined as follows. f < g if \{x\in A:f(x)<g(x)\}\in D. pcf(A) is the set of cofinalities that occur if we consider all ultrafilters on A, that is,

{\rm pcf}(A)=\{cf(\prod A/D):D\,\,\mbox{is an ultrafilter on}\,\,A\}.

Main results

Obviously, pcf(A) consists of regular cardinals. Considering ultrafilters concentrated on elements of A, we get that A\subseteq {\rm pcf}(A). Shelah proved, that if | A | < min(A), then pcf(A) has a largest element, and there are subsets \{B_\theta:\theta\in {\rm pcf}(A)\} of A such that for each ultrafilter D on A, cf(\prod A/D) is the least element θ of pcf(A) such that B_\theta\in D. Consequently, |{\rm pcf}(A)|\leq2^{|A|}. 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 ℵωis strong limit.

If λ is an infinite cardinal, then J is the following ideal on A. BJ if cf(\prod A/D)<\lambda holds for every ultrafilter D with BD. Then J is the ideal generated by the sets \{B_\theta:\theta\in {\rm pcf}(A),\theta<\lambda\}. There exist scales, i.e., for every λ∈pcf(A) there is a sequence of length λ of elements of \prod B_\lambda which is both increasing and cofinal mod J. This implies that the cofinality of \prod A under pointwise dominance is max(pcf(A)). Another consequence is that if λ is singular and no regular cardinal less than λ is Jónsson, then also λ+ is not Jónsson. In particular, there is a Jónsson algebra on ℵω+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 |A|<min(A). This would imply that if ℵω is strong limit, then the sharp bound


holds. The analogous bound


follows from Chang's conjecture (Magidor) or even from the nonexistence of a Kurepa tree (Shelah).

A weaker, still unsolved conjecture states that if |A|<min(A), then pcf(A) has no inaccessible limit point. This is equivalent to the statement that pcf(pcf(A))=pcf(A).


The theory has found a great deal of applications, besides cardinal arithmetic. The original survey by Shelah, Cardinal arithmetic for skeptics, includes the following topics: almost free abelian groups, partition problems, failure of preservation of chain conditions in Boolean algebras under products, existence of Jónsson algebras, existence of entangled linear orders, equivalently narrow Boolean algebras, and the existence of nonisomorphic models equivalent in certain infinitary logics.

In the meantime, many further applications have been found in Set Theory, Model Theory, Algebra and Topology.


Saharon Shelah, Cardinal Arithmetic, Oxford Logic Guides, vol. 29. Oxford University Press, 1994.

External links

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • PCF — may refer to: Contents 1 Computing 2 File formats 3 Technology 4 Other Computing Point coordination function …   Wikipedia

  • List of mathematics articles (P) — NOTOC P P = NP problem P adic analysis P adic number P adic order P compact group P group P² irreducible P Laplacian P matrix P rep P value P vector P y method Pacific Journal of Mathematics Package merge algorithm Packed storage matrix Packing… …   Wikipedia

  • Easton's theorem — In set theory, Easton s theorem is a result on the possible cardinal numbers of powersets. W. B. harvtxt|Easton|1970 (extending a result of Robert M. Solovay) showed via forcing that : kappa < operatorname{cf}(2^kappa),and, for kappale lambda,,… …   Wikipedia

  • List of unsolved problems in mathematics — This article lists some unsolved problems in mathematics. See individual articles for details and sources. Contents 1 Millennium Prize Problems 2 Other still unsolved problems 2.1 Additive number theory …   Wikipedia

  • Saharon Shelah — Infobox Scientist name = Saharon Shelah image width = 300px caption = Professor Saharon Shelah in Jerusalem, March 2008 birth date = Birth date and age|1945|7|3 birth place = Jerusalem residence = Jerusalem, Israel nationality = ISR ethnicity =… …   Wikipedia

  • Dowker space — A Dowker space is a topological space that is T4 but not countably paracompact. Equivalences If X is a normal T1 space (a T4 space), then the following are equivalent: X is a Dowker space The product of X with the unit interval is not normal. C.… …   Wikipedia

  • Charles Figley — is a university professor in the fields of psychology, family studies, social work, traumatology, and mental health. He is the Paul Henry Kurzweg, MD Distinguished Chair in Disaster Mental Health and Graduate School of Social Work Professor at… …   Wikipedia

  • Gimel function — In axiomatic set theory, the gimel function is the following function mapping cardinal numbers to cardinal numbers::gimelcolonkappamapstokappa^{mathrm{cf}(kappa)}where cf denotes the cofinality function; the gimel function is used for studying… …   Wikipedia

  • Fred Galvin — Frederick William Galvin is a mathematician, currently a professor at the University of Kansas. His research interests include set theory and combinatorics. His notable combinatorial work includes the proof of the Dinitz conjecture. In set theory …   Wikipedia

  • Cardinal function — In mathematics, a cardinal function (or cardinal invariant) is a function that returns cardinal numbers. Contents 1 Cardinal functions in set theory 2 Cardinal functions in topology 2.1 Basic inequalities …   Wikipedia