Definition:Ordinal Addition

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:


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

Also see

 * Ordinals under Addition form Ordered Semigroup