In the following, addition, multiplication and exponentiation of ordinals are defined using Transfinite Recursion.
For any ordinal $\alpha$:
- \[\alpha+0 = \alpha.\]
For all ordinals $\beta$,
\[\alpha+(\beta+1) = (\alpha+\beta)+1.\]For limit ordinals $\beta$,
\[\alpha+\beta = \lim_{\xi\to\beta}(\alpha+\xi).\]
For any ordinal $\alpha$:
- \[\alpha\cdot 0 = \alpha.\]
For all ordinals $\beta$,
\[\alpha\cdot(\beta+1) = \alpha\cdot\beta+\alpha.\]For limit ordinals $\beta$,
\[\alpha\cdot\beta = \lim_{\xi\to\beta}\alpha\cdot\xi.\]
For any ordinal $\alpha$:
- \[\alpha^0 = 1.\]
For all ordinals $\beta$,
\[\alpha^{\beta+1} = \alpha^\beta\cdot\alpha.\]For limit ordinals $\beta$,
\[\alpha^\beta = \lim_{\xi\to\beta}\alpha^\xi.\]
OAR#PROP-ADD-ASS. Associativity of $+$.
For any ordinals $\alpha$, $\beta$ and $\gamma$,
\[(\alpha+\beta)+\gamma = \alpha+(\beta+\gamma).\]
OAR#PROP-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:
\[1+\omega=\omega\neq\omega+1, \quad 2\cdot\omega=\omega\neq\omega\cdot 2.\]