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 \(S\).[1]p.258 Cofinality of \(\alpha\) is denoted \(\mathrm{cf} \alpha\)[1]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 \(\alpha\).[1]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 \(\lambda\).[1]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[]
- ↑ 1.0 1.1 1.2 1.3 Herbert Enderton, Elements of Set Theory, Elsevier Science, 1977.