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glossary 
COFINALITY AND

Cofinality
The COFINALITY of a wellordered 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 wellordered set X, cf(X) is a cardinal.
 cf is idempotent, i.e., cf(cf(X)) = cf(X).
 For any wellordered 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., aleph_{0} 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.
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Nancy McGough Last significant update on May 23, 1997 Last tweak on May 31, 1997 
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