cofinalities

Cofinality is a set‑theoretic invariant that measures how a partially ordered set, most commonly an ordinal or a cardinal, can be approached from below. For an ordinal α, the cofinality cf(α) is defined as the smallest cardinality of a subset S⊆α that is unbounded in α; that is, for every β<α there exists s∈S with β≤s. Thus cf(α) reflects the least length of an increasing sequence that converges to α in the order topology.

If α is a successor ordinal, then cf(α)=1, because the singleton consisting of its immediate predecessor is already cofinal. For a limit ordinal, cofinality can be finite or infinite. Classic examples include cf(ω)=ω and cf(ω₁)=ω₁, the latter showing that ω₁ is a regular cardinal. A cardinal κ is called regular when cf(κ)=κ and singular when cf(κ)<κ; singular cardinals include ℵ_ω, whose cofinality is ω.

Cofinality behaves well with respect to order isomorphisms, so equivalent ordinals share the same cofinality. It is also preserved under taking successor cardinals: cf(κ⁺)=κ⁺. In general, the cofinality of an infinite cardinal is a regular cardinal smaller than or equal to the cardinal itself.

Beyond ordinals, cofinality can be defined for arbitrary partially ordered sets: it is the minimum cardinality of a cofinal substructure. This concept underlies many results in combinatorial set theory, such as the singular cardinal theorem and the study of large cardinals. Cohesion with cardinal arithmetic leads to conjectures such as the singular cardinal hypothesis, which concerns how the exponentiation of singular cardinals behaves relative to their cofinalities.