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This paper mentions that any ordinal $\alpha < \epsilon_0$ can be written uniquely as $\alpha = \omega^\beta(\gamma+1)$, where $\beta<\alpha$. Does this presentation have a name? Also, how would one get $\omega^\beta(\gamma+1)$ for a given $\alpha$? I played with this for a while and got stuck when setting $\alpha$ to, say, $\omega^\omega+1$. (Or is this just Cantor normal form disguised?) |
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This is very close to the Cantor Normal Form, and likely has no name of its own. Note that given $\alpha < \varepsilon_0$ its Cantor Normal Form is $$\omega^{\beta_1} \cdot k_1 + \cdots + \omega^{\beta_n} \cdot k_n$$ where $\alpha > \beta_1 > \cdots > \beta_n$ and $0 < k_1 , \ldots , k_n < \omega$. Now essentially factor out $\omega^{\beta_n}$ to the left to get the representation (as $0 < k_n < \omega$ whatever is left over after factoring out must be a successor ordinal). |
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