## Definition

Let $x$ and $y$ be ordinals.

The operation of ordinal addition $x + y$ is defined using the Second Principle of 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:

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

## Examples

Let $x$ be an ordinal.

Let $x^+$ denote the successor of $x$.

Let $1$ denote (ordinal) one, the successor of the zero ordinal $\O$.

Then:

$x + 1 = x^+$

where $+$ denotes ordinal addition.

Let $2$ denote the successor of the ordinal $1$.

Then:

$x + 2 = x^{++}$

### Ordinal Addition by Natural Number

Let $n$ be a natural number.

Then:

$x + \paren {n + 1} = \paren {x + n}^+$