cofinality

Cofinality is a notion from order theory and set theory that measures how large a subset must be to approximate the top end of a partially ordered set. For a nonempty poset P, the cofinality cf(P) is the least cardinal κ for which there exists a cofinal subset A ⊆ P of size κ; a subset A is cofinal if for every p ∈ P there exists a ∈ A with p ≤ a.

In the standard cases of ordinals and cardinals, cf has explicit meanings. For an ordinal α, cf(α) is the least β such that α is the limit of an increasing sequence of length β with values below α. Consequently cf(0)=0, cf(1)=1, and if α is a successor then cf(α)=1. For a limit ordinal α, cf(α) is the minimal length of a cofinal increasing sequence approaching α. For a cardinal κ, cf(κ) is computed by viewing κ as the initial ordinal; κ is called regular when cf(κ)=κ, and singular when cf(κ)<κ. Examples: cf(ω)=ω (ω is regular), cf(ω1)=ω1 (ω1 is regular), cf(aleph_ω)=ω (aleph_ω is singular).

Cofinality is a central tool in set theory and model theory. It constrains how cardinals can be combined and is involved in results such as König’s theorem, which relates cofinalities of products and sums of cardinals. It also explains why certain infinite objects must have inherent long cofinal sequences, and it is used to analyze forcing notions and the structure of large cardinals.