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 $y = \O$, define:


 * $x + \O := x$

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


 * $x + y^+ := \paren {x + y}^+$

Limit Case
Let $y$ be a limit ordinal. Then:


 * $\ds x + y := \bigcup_{z \mathop \in y} \paren {x + z}$

Also see

 * Ordinals under Addition form Ordered Semigroup