Cofinality

The cofinality of an ordinal \(\alpha\), denoted \(\mathrm{cof}(\alpha)\) or \(\mathrm{cf}(\alpha)\), is the least \(\mu\) so that there is some function \(f: \mu \to \alpha\) with unbounded range. For example:

• The cofinality of \(0\) is \(0\).
• The cofinality of any successor ordinal is \(1\), because the map \(f: 1 \to \alpha+1\) defined by \(f(0) = \alpha\) has unbounded range.
• The cofinality of any limit of ordinal is at least \(\omega\): if it's countable, then it's exactly \(\omega\).

It is easy to see that \(\mathrm{cof}(\alpha) \leq \alpha\) for all \(\alpha\), because the identity has unbounded range. Also, \(\mathrm{cof}(\mathrm{cof}(\alpha)) = \mathrm{cof}(\alpha)\), because if there is a \(\delta < \mathrm{cof}(\alpha)\) and maps \(f: \delta \to \mathrm{cof}(\alpha)\), \(g: \mathrm{cof}(\alpha) \to \alpha\) with unbounded range, then \(g \circ f: \delta \to \alpha\) also has unbounded range, contradicting minimality of \(\mathrm{cof}(\alpha)\).

An ordinal is regular if it is equal to its own cofinality, else it is singular. So:

• 0, 1 and \(\omega\) are regular.
• All natural numbers other than \(1\) are singular.
• All countable ordinals other than \(\omega\) are singular.
• \(\mathrm{cof}(\alpha)\) is regular for any \(\alpha\).

Cofinality is used in the definition of weakly inaccessible cardinals.

Without choice

Citation about every uncountable cardinal being singular being consistent with ZF