Definition:Cofinality

Formal Definition

Let $\kappa$ be an ordinal.

The cofinality of $\kappa$, denoted $\operatorname{cf} \left({ \kappa }\right)$, is the smallest ordinal $\lambda$ such that:

$\operatorname{cof} \left({ \kappa , \lambda }\right)$

where $\operatorname{cof} \left({ \kappa , \lambda }\right)$ denotes the cofinal relation.

Intuitive Definition

Cofinality describes the smallest ordinal capable of reaching $\kappa$ "from below" using some mapping.