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

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

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


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