Climb into Cantor’s Attic, where you will find infinities large and small. We aim to provide a comprehensive resource of information about all notions of mathematical infinity.
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A limit ordinal is an ordinal that is neither $0$ nor a successor ordinal. Some authors classify zero as a limit ordinal.
All limit ordinals are equal to their union.
All limit ordinals contain an ordinal $\alpha$ if and only if they contain $\alpha + 1$.
$\omega$ is the smallest nonzero limit ordinal, and the smallest ordinal of infinite cardinal number.
$(\omega + \omega)$, also written $( \omega \cdot 2 )$, is the next limit ordinal. $( \omega \cdot \alpha )$ is a limit ordinal for any ordinal $\alpha$.
A limit ordinal $\alpha$ is called additively indecomposable (or a $\gamma$ number) if it cannot be the sum of $\beta<\alpha$ ordinals less than $\alpha$. These numbers are any ordinal of the form $\omega^\beta$ for $\beta$ an ordinal. The smallest is written $\gamma_0$, and the smallest larger than that is $\gamma_1$, etc.
A limit ordinal $\alpha$ is called multiplicatively indecomposable (or a $\delta$ number) if it cannot be the product of $\beta<\alpha$ ordinals less than $\alpha$. These numbers are any ordinal of the form $\omega^{\omega^{\beta}}$. The smallest is written $\delta_0$, and the smallest larger than that is $\delta_1$, etc.
Interestingly, this pattern does not continue with exponentially indecomposable (or $\varepsilon$ numbers) ordinals being $\omega^{\omega^{\omega^\beta}}$, but rather $\varepsilon_0=sup_{n<\omega}f^n(0)$ with $f(\alpha)=\omega^\alpha$ and $f^n(\alpha)=f(f(…f(\alpha)…))$ with $n$ iterations of $f$. It is the smallest fixed point of $f$. The next $\varepsilon$ number (i.e. the next fixed point of $f$) is then $\varepsilon_1=sup_{n<\omega}f^n(\varepsilon_0+1)$, and more generally the $(\alpha+1)$th fixed point of $f$ is $\varepsilon_{\alpha+1}=sup_{n<\omega}f^n(\varepsilon_\alpha+1)$, also $\varepsilon_\lambda=\cup_{\alpha<\lambda}\varepsilon_\alpha$ for limit $\lambda$.
The tetrationally indecomposable ordinals (or $\zeta$ numbers) are then the ordinals $\zeta$ such that $\varepsilon_\zeta=\zeta$. These are obtained similarly as $\varepsilon$ numbers by taking $f(\alpha)=\varepsilon_\alpha$. Pentationally indecomposable ordinals (or $\eta$ ordinals) are then obtained by taking $f(\alpha)=\zeta_\alpha$, and so on.
This pattern continues on with the Veblen Hiearchy, continuing up to the Feferman-Schütte ordinal $\Gamma_0$, the smallest ordinal such that this process does not generate any larger kind of ordinals.