In the following, addition, multiplication and exponentiation of ordinals are defined using Transfinite Recursion.
For any ordinal $\alpha$,
- \[\alpha+0 = \alpha;\]
- \[\alpha+(\beta+1) = (\alpha+\beta)+1\]
for non-limit ordinals $\beta$;
- \[\alpha+\beta = \lim_{\xi\to\beta}(\alpha+\xi)\]
for limit ordinals $\beta$.
For any ordinal $\alpha$,
- \[\alpha\cdot 0 = \alpha;\]
- \[\alpha\cdot(\beta+1) = \alpha\cdot\beta+\alpha\]
for non-limit ordinals $\beta$;
- \[\alpha\cdot\beta = \lim_{\xi\to\beta}\alpha\cdot\xi\]
for limit ordinals $\beta$.
For any ordinal $\alpha$,
- \[\alpha^0 = 1;\]
- \[\alpha^{\beta+1} = \alpha^\beta\cdot\alpha\]
for non-limit ordinals $\beta$;
- \[\alpha^\beta = \lim_{\xi\to\beta}\alpha^\xi\]
for limit ordinals $\beta$.
PROP-OAR-ADD-ASS. Associativity of $+$.
For any ordinals $\alpha$, $\beta$ and $\gamma$,
\[\alpha+(\beta+\gamma) = (\alpha+\beta)+\gamma.\]
PROP-OAR-MUL-ASS. Associativity of $\cdot$.
For any ordinals $\alpha$, $\beta$ and $\gamma$,
\[\alpha\cdot(\beta\cdot\gamma) = (\alpha\cdot\beta)\cdot\gamma.\]
Neither $+$ or $\cdot$ is commutative:
\[1+\omega=\omega\neq\omega+1, \quad 2\cdot\omega=\omega\neq\omega\cdot 2.\]