Cofinality

The cofinality of any ordinal \(\alpha\) is the least cardinal number \(\kappa\) such that there exists a subset \(S\) of \(\alpha\) having cardinality \(\kappa\) and \(\alpha\) is the least ordinal strictly greater than every member of \(S\)., Elements of Set Theory, Elsevier Science, 1977.p.258 Cofinality of \(\alpha\) is denoted \(\mathrm{cf} \alpha\)p.258 or \(\mathrm{cf}(\alpha)\).
For example, the cofinality of \(\omega2\) is \(\omega\) because
* a subset \(S = \{\omega + n|n \in \mathbb{N}\}\) has such property; cardinality is \(\omega\) and \(\omega2\) is the least ordinal strictly greater than every member of \(S\).
* for a subset \(S\) having non-zero finite cardinality, there is the largest member \(\beta\), and \(\beta +1\) is strictly greater than every member of \(S\) while smaller than \(\omega2\).
* for the case \(S = \emptyset\), \(0\) is strictly greater than every member of \(S\) while smaller than \(\omega2\).
The cofinality of any countable limit ordinal \(\alpha\) is ω, because \(S = \alpha\) satisfies the desired property. The cofinality of any successor ordinal \(\beta+1\) is 1, because \(S = \{\beta\}\) satisfies the desired property. The cofinality of 0 is 0, because \(S = \emptyset\) satisfies the desired property.
== Characterizing cofinality with increasing sequence ==
For an ordinal number \(\alpha\), define an \(\alpha\)-sequence to be a function with domain \(\alpha\).p.258 For example, an ordinary infinite sequence is an \(\omega\)-sequence, and a finite sequence is an \(n\)-sequence for some natural number \(n\).
Assume that \(\lambda\) is a limit ordinal. Then there is an increasing (\(\textrm{cf}(\lambda)\))-sequence into \(\lambda\) that converges to \(\lambda\).p.259 Theorem 9Q When \(\textrm{cf}(\lambda) = \omega\), the sequence may also be called a fundamental sequence.
== Regular ==
An ordinal is called regular if its cofinality equals itself.
== Sources ==
Category:Transfinite ordinals
ru:Кофинальность