Cantor normal form

Every nonzero ordinal has a unique finite expression ${\omega }^{{\delta }_1}{n}_1+\cdots +{\omega }^{{\delta }_r}{n}_r$ with decreasing exponents and positive finite coefficients.

Finite canonical terms built from zero, ordinal sums and powers of omega describe exactly the ordinals below epsilon zero. Syntactic validity and comparison are decidable by recursion through the exponent subterms. Enumerating valid term codes transfers their order to a computable well-order on all natural numbers.

The leading term ${\omega }^{\delta }n$ is the highest-exponent term in Cantor normal form; the remaining tail is strictly below ${\omega }^{\delta }$.

Every nonzero ordinal $\alpha$ has a greatest exponent $\beta$ such that ${\omega }^{\beta }\le \alpha$. It is the leading exponent in the Cantor normal form of $\alpha$, andfor a positive finite integer $n$ and $\gamma <{\omega }^{\beta }$.

Let nonzero ordinals $\xi ,\eta$ have leading exponents $\rho ,\sigma$. If $\eta$ is finite, then $\xi \eta$ has leading exponent $\rho$. If $\eta$ is infinite, so that $\sigma >0$, then $\xi \eta$ has leading exponent $\rho +\sigma$. This follows by multiplying the leading terms in Cantor normal form and using continuity of ordinal multiplication at limit ordinals.