Cofinality of a reflexive order ≼. This is the smallest cardinality of a subset S : set α such that ∀ a, ∃ b ∈ S, a ≼ b.

Cofinality of an ordinal. This is the smallest cardinal of a subset S of the ordinal which is unbounded, in the sense ∀ a, ∃ b ∈ S, ¬(b > a). It is defined for all ordinals, but cof 0 = 0 and cof (succ o) = 1, so it is only really interesting on limit ordinals (when it is an infinite cardinal).

If the union of s is unbounded and s is smaller than the cofinality, then s has an unbounded member

If the union of s is unbounded and s is smaller than the cofinality, then s has an unbounded member

The infinite pigeonhole principle

pigeonhole principle for a cardinality below the cardinality of the domain

A cardinal is a limit if it is not zero or a successor cardinal. Note that ω is a limit cardinal by this definition.

A cardinal is a strong limit if it is not zero and it is closed under powersets. Note that ω is a strong limit by this definition.

A cardinal is regular if it is infinite and it equals its own cofinality.

A cardinal is inaccessible if it is an uncountable regular strong limit cardinal.