Trapped in a frame?
Break free now!

glossary

COFINALITY AND REGULAR & SINGULAR ORDINALS

Cofinality

The COFINALITY of a well-ordered set X, denoted cf(X), is the least ordinal a such that a can be mapped unboundedly into X. Cofinality gives a measure of how "reachable from below" an ordinal is.

Cofinality Examples

• cf(aleph_w) = w because the mapping f:w->aleph_w defined by f(n) = aleph_n is unbounded in aleph_w.
• If X is a successor ordinal, i.e., X=a+1={0, 1, ..., a}, then cf(X)=1 because the mapping f:1->X, i.e. f:{0}->{0, 1, ..., a}, defined by f(0)=a is unbounded in X.

Facts About Cofinality

• For any well-ordered set X, cf(X) is a cardinal.
• cf is idempotent, i.e., cf(cf(X)) = cf(X).
• For any well-ordered set X of order type a, cf(X) <= a. In particular, for any ordinal a, cf(a) <= a.

Regular and Singular Ordinals

The ordinal a is said to be REGULAR if cf(a) = a and SINGULAR if cf(a) < a.

• If a is regular then a is a cardinal.
• In ZF, it is unknown whether one can prove that there exists a cardinal of cofinality > w.
• In ZFC, it is not possible to prove that there is an uncountable regular limit cardinal. I.e., aleph₀ is the only infinite regular limit cardinal that can be `constructed' using the axioms of ZFC.

See Also

Cardinal numbers especially regular, singular, weakly_inaccessible, and inaccessible cardinals.

top of this page about ii ... directory ... getting in touch ... we're looking for ... ... bookstore ... linking & mirrors ... legal notices ... thanks!

Copyright © 1992-1997 Infinite Ink and Nancy McGough Last significant update on May 23, 1997 Last tweak on May 31, 1997	www.ii.com/glossary/cofinality_regular_singular/ www.best.com/~ii/glossary/cofinality_regular_singular/