# Definition:Cofinality

## Formal Definition

Let $\kappa$ be an ordinal.

The cofinality of $\kappa$, denoted $\map {\mathrm {cf} } \kappa$, is the smallest ordinal $\lambda$ such that:

$\map {\mathrm {cof} } {\kappa, \lambda}$

where $\map {\mathrm {cof} } {\kappa, \lambda}$ denotes the cofinal relation.

## Intuitive Definition

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