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glossary

# COFINALITY ANDREGULAR & 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(alephw) = w because the mapping f:w->alephw defined by f(n) = alephn is unbounded in alephw.
• 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.

• 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.