Definition:Ordinal Addition

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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