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.

For every ordinal base $\rho >1$, a nonzero ordinal has a unique finite ordinal addition expansion with strictly decreasing exponents ${\beta }_i$ and coefficients $0<{\xi }_i<\rho$. Choose the largest power at most the remaining ordinal and apply the ordinal division algorithm. The next remainder has smaller leading exponent, so well-foundedness forces termination. Each lower tail is smaller than its preceding power; uniqueness of division consequently fixes the exponent, coefficient and tail successively. For $\rho =\omega$ this is Cantor normal form.

Two ordinals are commensurable when neither is absorbed by adding the other on its left. For positive ordinals, this is equivalent to equality of their leading Cantor normal form exponent. If leading exponents differ, the lower-leading ordinal is absorbed by the higher power of $\omega$. If they agree, either sum adds the two positive leading coefficients and therefore exceeds both original ordinals. Zero is commensurable with no ordinal.

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.

Use the unary Cantor normal form tree norm, counting one root and recursively counting all repeated exponent subterms. There are only finitely many notations of bounded norm. This finite principle bounds all descending sequences whose norms grow at most linearly with the position. Friedman's finite form of Kruskal's theorem implies it through the natural-sum ordinal rank of a finite rooted tree. The Friedman–Smith miniaturization theorem turns this controlled termination principle for standard epsilon-zero notations into the formal consistency statement for Peano arithmetic; the technical ordinal-analysis result is needed in addition to the tree coding. Section 2.2 of formal.hknu.ac.kr/Publi/kruskalJKMS.pdf records the general miniaturization and soundness theorem.

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.