A limit ordinal is an [ordinal](Ordinal.md "Ordinal") that is neither [$0$](Zero.md "Zero") nor a [](Successor%20ordinal.md). Some authors classify zero as a limit ordinal. ## Properties 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](Cardinal.md "Cardinal") number. $(\omega + \omega)$, also written $( \omega \cdot 2 )$, is the next limit ordinal. $( \omega \cdot \alpha )$ is a limit ordinal for any ordinal $\alpha$. ## Types of Limits 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 [](Feferman-Schütte.md), continuing up to the [Feferman-Schütte](Feferman-Schütte.md "Feferman-Schütte") ordinal $\Gamma_0$, the smallest ordinal such that this process does not generate any larger kind of ordinals.