Definition 1. For any ordinal $\alpha$:
- \[\alpha + 0 = \alpha.\]
-
For all ordinals $\beta$,
\[\alpha + (\beta + 1) = (\alpha + \beta) + 1.\] -
For limit ordinals $\beta$,
\[\alpha + \beta = \lim_{\xi \to \beta} (\alpha + \xi).\]
In the following, addition, multiplication and exponentiation of ordinals are defined using Transfinite Recursion.
Definition 2. For any ordinal $\alpha$:
- \[\alpha\cdot 0 = \alpha.\]
-
For all ordinals $\beta$,
\[\alpha\cdot(\beta+1) = \alpha\cdot\beta+\alpha.\] -
For limit ordinals $\beta$,
\[\alpha\cdot\beta = \lim_{\xi\to\beta}\alpha\cdot\xi.\]
Definition 3. For any ordinal $\alpha$:
- \[\alpha^0 = 1.\]
-
For all ordinals $\beta$,
\[\alpha^{\beta + 1} = \alpha^\beta \cdot \alpha.\] -
For limit ordinals $\beta$,
\[\alpha^\beta = \lim_{\xi \to \beta} \alpha ^ \xi.\]
Proposition 1 (Associativity of $+$). For any ordinals $\alpha$, $\beta$ and $\gamma$,
\[(\alpha + \beta) + \gamma = \alpha + (\beta + \gamma).\]Proposition 2 (Associativity of $\cdot$). For any ordinals $\alpha$, $\beta$ and $\gamma$,
\[(\alpha \cdot \beta) \cdot \gamma = \alpha \cdot (\beta \cdot \gamma).\]Remark 1. Neither $+$ or $\cdot$ is commutative:
\[1 + \omega = \omega \neq \omega + 1, \quad% 2 \cdot \omega = \omega \neq \omega \cdot 2.\]