Definition:Ordinal Addition

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

Ordinal Addition by One

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.

Ordinal Addition by Two

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:

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

Also see

• Ordinals under Addition form Ordered Semigroup

• Results about ordinal addition can be found here.

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 8.1$