# Definition:Cofinal Relation on Ordinals

## Contents

## Definition

Let $x$ and $y$ be ordinals.

Then $y$ is said to be **cofinal** with respect to $x$ if and only if there exists a mapping $f: y \to x$ such that:

- $(1): \quad y \le x$
- $(2): \quad f$ is strictly increasing.
- $(3): \quad$ For all $a \in x$, there is some $b \in y$ such that $f \left({b}\right) \ge a$.

## Notation

If $y$ is **cofinal** with $x$, the notation $\operatorname{cof} \left({x, y}\right)$ can be used.

## Warning

$\operatorname{cof}$ is not symmetric. In fact, it is antisymmetric.

## Also see

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 10.51$