Ordinal Arithmetic

Ordinal arithmetic extends addition, multiplication, and exponentiation to ordinal numbers using transfinite recursion. Unlike cardinal arithmetic, ordinal operations are sensitive to order and generally non-commutative.

Definitions by Transfinite Recursion

Definition4.4Ordinal addition

For ordinals $\alpha, \beta$ , ordinal addition $\alpha + \beta$ is defined by recursion on $\beta$ :

$\alpha + 0 = \alpha$ .
$\alpha + (\beta + 1) = (\alpha + \beta) + 1$ .
$\alpha + \lambda = \sup\{\alpha + \beta : \beta < \lambda\}$ for limit $\lambda$ .

Set-theoretically, $\alpha + \beta$ is the order type of $(\alpha \times \{0\}) \cup (\beta \times \{1\})$ with lexicographic order on the second component.

Definition4.5Ordinal multiplication

Ordinal multiplication $\alpha \cdot \beta$ is defined by recursion on $\beta$ :

$\alpha \cdot 0 = 0$ .
$\alpha \cdot (\beta + 1) = \alpha \cdot \beta + \alpha$ .
$\alpha \cdot \lambda = \sup\{\alpha \cdot \beta : \beta < \lambda\}$ for limit $\lambda$ .

Set-theoretically, $\alpha \cdot \beta$ is the order type of $\beta \times \alpha$ with reverse lexicographic order.

Definition4.6Ordinal exponentiation

Ordinal exponentiation $\alpha^\beta$ is defined by recursion on $\beta$ :

$\alpha^0 = 1$ .
$\alpha^{\beta+1} = \alpha^\beta \cdot \alpha$ .
$\alpha^\lambda = \sup\{\alpha^\beta : \beta < \lambda\}$ for limit $\lambda$ (when $\alpha > 0$ ).

Non-Commutativity and Examples

ExampleOrdinal arithmetic computations

Addition: $\omega + 1 \neq 1 + \omega = \omega$ . Also $\omega + \omega = \omega \cdot 2$ , but $2 + \omega = \omega$ .

Multiplication: $\omega \cdot 2 = \omega + \omega$ , but $2 \cdot \omega = \sup\{2 \cdot n : n < \omega\} = \sup\{0, 2, 4, 6, \ldots\} = \omega$ . So $\omega \cdot 2 \neq 2 \cdot \omega$ .

Exponentiation: $2^\omega = \sup\{2^n : n < \omega\} = \omega$ , but $\omega^2 = \omega \cdot \omega$ . And $\omega^\omega$ is the order type of the set of finite sequences from $\omega$ ordered anti-lexicographically.

Cantor Normal Form

Theorem4.2Cantor normal form

Every ordinal $\alpha > 0$ can be uniquely written as:

$\alpha = \omega^{\beta_1} \cdot c_1 + \omega^{\beta_2} \cdot c_2 + \cdots + \omega^{\beta_k} \cdot c_k,$

where $k \geq 1$ , $\alpha \geq \beta_1 > \beta_2 > \cdots > \beta_k \geq 0$ , and $c_1, \ldots, c_k$ are positive finite integers. This is the Cantor normal form of $\alpha$ .

ExampleCantor normal form examples

$\omega^2 \cdot 3 + \omega \cdot 5 + 7$ : already in Cantor normal form.
$\omega^\omega + \omega^3 \cdot 2 + 1$ : Cantor normal form with exponents $\omega > 3 > 0$ .
$\varepsilon_0 = \omega^{\omega^{\omega^{\cdot^{\cdot^{\cdot}}}}}$ : the Cantor normal form of $\varepsilon_0$ is $\omega^{\varepsilon_0} \cdot 1$ , since $\omega^{\varepsilon_0} = \varepsilon_0$ .

RemarkLeft cancellation and absorption

Ordinal addition satisfies left cancellation: $\alpha + \beta = \alpha + \gamma \implies \beta = \gamma$ . However, right cancellation fails: $1 + \omega = 0 + \omega$ but $1 \neq 0$ . The "absorption" $n + \omega = \omega$ for finite $n$ reflects the fact that finitely many elements before $\omega$ -many are "absorbed" into the limit.