nLab
cofinality

Cofinality

Idea

The cofinality of a quoset (quasi-ordered set) is a measure of the size of the quoset and in particular of the size of its tails. An important special case is the cofinality of an ordinal number, and there is a related concept of the cofinality of a cardinal number.

Definitions

We begin with definitions that work even in weak foundations of mathematics.

Given a quasi-ordered set QQ, the cofinality of QQ is the collection of all cardinal numbers κ\kappa such that every function f:[κ]Qf\colon [\kappa] \to Q (where [κ][\kappa] is any set of cardinality κ\kappa) has a (strict) upper bound: an element xx of QQ such that, whenever yy belongs to the image of ff, y<xy \lt x. A priori, this collection Cf(Q)Cf(Q) may be a proper class, but it is often a set, indeed always in classical mathematics (as shown below). We traditionally write κ<cf(Q)\kappa \lt cf(Q) to mean κCf(Q)\kappa \in Cf(Q) (for reasons to be seen below).

The ordinal cofinality of QQ is the collection Ocf(Q)Ocf(Q) of all ordinal numbers α\alpha such that |α|<cf(Q){|\alpha|} \lt cf(Q). This collection is clearly a down-set and so may be identified with an ordinal number ocf(Q)\ocf(Q), also called the ordinal cofinality; so we may write α<ocf(Q)\alpha \lt ocf(Q) in place of αOcf(Q)\alpha \in Ocf(Q), although traditionally we simply write α<cf(Q)\alpha \lt cf(Q).

If we start with a collection CC of cardinal numbers, the cardinal cofinality of CC is the collection Ccf(C)Ccf(C) of all cardinal numbers κ\kappa such that, given any [κ][\kappa]-indexed family FF of sets, each of which has cardinality in CC, the disjoint union of this family (or equivalently the union in a material set theory) also has cardinality in CC. Again we write κ<ccf(C)\kappa \lt ccf(C) or even κ<cf(C)\kappa \lt cf(C) to mean κCcf(C)\kappa \in Ccf(C).

Identifications

Assume the axiom of choice. Then we may identify and simplify some of the concepts above.

  • As a class of cardinal numbers, cf(Q)cf(Q) is clearly a down-set (that is closed under subsets), so it must be the set {κ|κ<cf(Q)}\{\kappa \;|\; \kappa \lt cf(Q)\} for some cardinal number cf(Q)cf(Q), also called the cofinality. (Note that cf(Q)|Q|cf(Q) \leq {|Q|}, equivalently |Q|cf(Q){|Q|} \nless cf(Q), since the identity function QQQ \to Q has no upper bound, so in particular we are not dealing with proper classes.) In this case, we conclude that there is a function [cf(Q)]Q[cf(Q)] \to Q that has no strong upper bound, and that cf(Q)cf(Q) is the smallest cardinal number with this property, which is the usual definition. Assuming that QQ is a linear order, it follows that the image of some function [cf(Q)]Q[cf(Q)] \to Q is cofinal? in QQ (whence the terminology).

  • Using the identification of cardinal numbers with certain von Neumann ordinals, the ordinal cofinality Ocf(Q)Ocf(Q) or ocf(Q)ocf(Q) becomes identified with the classical cofinality cf(Q)cf(Q). (But note that Cf(Q)Cf(Q), the collection of cardinal numbers, is only a subset of Ocf(Q)Ocf(Q) when we identify cardinals as certain ordinals.)

  • Every cardinal cofinality Ccf(C)Ccf(C) is also a down-set of cardinal numbers, hence of the form {κ|κ<ccf(C)}\{\kappa \;|\; \kappa \lt ccf(C)\} for some cardinal number ccf(C)ccf(C). Furthermore, if we start with a cardinal number λ\lambda and define the collection C λ{κ|κ<λ}C_\lambda \coloneqq \{\kappa \;|\; \kappa \lt \lambda\} of cardinal numbers, then cf([λ])=ccf(C λ)cf([\lambda]) = ccf(C_\lambda). If we identify λ\lambda with a von Neumann ordinal, then we also have cf([λ])=ocf(λ)cf([\lambda]) = ocf(\lambda), so all notions of cofinality agree.

Properties

Here is an important theorem on ordinal cofinality, which following our definitions is entirely constructive:

ocf(ocf(Q))=ocf(Q). ocf(ocf(Q)) = ocf(Q) .

In general, an ordinal number α\alpha such that ocf(α)=αocf(\alpha) = \alpha is called regular, so every ordinal cofinality is regular. For example, 00, 11, and ω\omega are regular ordinals.

A regular cardinal may be defined to be a collection of cardinals CC such that Ccf(C)=CCcf(C) = C. Assuming the axiom of choice and making identifications as above, the regular cardinals and the regular ordinals are the same, except that 22 is a regular cardinal (but not a regular ordinal). Also, {1}\{1\} is a regular collection of cardinals that is not a down-set, although every other regular cardinal is (and so can be identified with a cardinal number), classically.

Traditionally, one requires a regular ordinal or cardinal to be infinite, and thus classically they are the same with no exceptions.

Revised on July 29, 2014 21:02:08 by Toby Bartels (98.19.42.133)