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.