## Definition

Let $x$ and $y$ be ordinals.

The operation of ordinal addition $x + y$ is defined using transfinite recursion on $y$, as follows.

### Base Case

When $y = \varnothing$, define:

$x + \varnothing := x$

### Inductive Case

For a successor ordinal $y^+$, define:

$x + y^+ := \left({x + y}\right)^+$

### Limit Case

Let $y$ be a limit ordinal. Then:

$\displaystyle x + y := \bigcup_{z \mathop \in y} \left({x + z}\right)$