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Ordinal exponentiation is defined byfor limit $\lambda$. Transfinite induction gives ${\omega }^{\alpha }\ge \alpha$. Since ${\omega }^{\alpha +1}>\alpha$, there is a least ${\alpha }^{\ast }$ with ${\omega }^{{\alpha }^{\ast }}>\alpha$. If nonzero ${\alpha }^{\ast }$ were a limit, the defining supremum would imply ${\omega }^{\beta }>\alpha$ for some $\beta <{\alpha }^{\ast }$, a contradiction. Hence ${\alpha }^{\ast }$ is a successor.

For $\alpha >0$, write ${\alpha }^{\ast }=\beta +1$. Then ${\omega }^{\beta }\le \alpha <{\omega }^{\beta +1}$. There is a largest positive integer $n$ with ${\omega }^{\beta }n\le \alpha$, and ordinal division givesRepeating on $\gamma$ terminates because there is no infinite strictly decreasing sequence of ordinals, producing the Cantor normal form

For two monomials,For general $\alpha +{\alpha }^{\prime }$, compare the leading exponent ${\beta }_1^{\prime }$ with the exponents of $\alpha$: discard every trailing term of $\alpha$ having exponent below ${\beta }_1^{\prime }$, combine coefficients if the last retained exponent equals ${\beta }_1^{\prime }$, and then append all remaining terms of ${\alpha }^{\prime }$. This is its Cantor normal form.

True. The hypothesis says that $\alpha +\alpha$ and $\beta$ commute under ordinal addition. By the commuting ordinal addition classification, comparison of their Cantor normal forms implies that $\alpha$ and $\beta$ themselves commute. Thereforeso both outer expressions equal the middle one.

First let $\delta ={\omega }^{\alpha }$. If $\beta ,\gamma <\delta$, every exponent appearing in their Cantor normal forms is strictly below $\alpha$. The rules for ordinal addition can delete lower terms but cannot introduce an exponent at least $\alpha$. Thereforeso $\delta$ is additively closed.

Conversely, suppose nonzero $\delta$ is additively closed. Part (a) givesIf $\rho >0$, then both ${\omega }^{\alpha }n$ and $\rho$ are smaller than $\delta$, but their sum is $\delta$, contradicting closure. Hence $\rho =0$. If now $n>1$, thenwhileanother contradiction. Thus $n=1$, andThis proves the classification of additively closed ordinals.

Suppose first that $\delta >2$ is multiplicatively closed. It is also additively closed. Indeed, for $\beta ,\gamma <\delta$, let $\mu =\max \{\beta ,\gamma \}$. Monotonicity of ordinal addition giveswhere the last inequality uses $\mu <\delta$, $2<\delta$, and multiplicative closure. Part (b) therefore givesfor some nonzero ordinal $\lambda$.

For any $\rho ,\sigma <\lambda$, strict monotonicity givesMultiplicative closure and the exponent law now implyhence $\rho +\sigma <\lambda$. Thus $\lambda$ is additively closed, and part (b) gives $\lambda ={\omega }^{\alpha }$. Consequently

Conversely, let $\delta ={\omega }^{\lambda }$ with $\lambda ={\omega }^{\alpha }$. By part (b), $\lambda$ is additively closed. Take nonzero $\beta ,\gamma <\delta$, with leading exponents $\rho ,\sigma <\lambda$ in Cantor normal form. If $\gamma$ is finite, the product $\beta \gamma$ has leading exponent $\rho <\lambda$. If $\gamma$ is infinite, the leading exponent of an ordinal product isby additive closure of $\lambda$. In either caseProducts involving zero are immediate, so $\delta$ is multiplicatively closed. This is the multiplicative closure criterion for a power of omega and completes both directions.

Write the nonzero ordinal in Cantor normal formwhere ${\delta }_1>\cdots >{\delta }_r$ and $0<n_i<\omega$. Taking $\delta ={\delta }_1$, $n=n_1$, andgives $\alpha ={\omega }^{\delta }n+\eta$ with $\eta <{\omega }^{\delta }$.

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.