Definition:Ordinal Addition

Definition
Let $x$ and $y$ be ordinals. We shall define $x+y$ using transfinite recursion.

Base Case

 * $\displaystyle x + \varnothing = x$

Inductive Case

 * $\displaystyle x + y^+ = (x+y)^+$

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


 * $\displaystyle x+y = \bigcup_{z \in y} (x+z)$