| Version 4 |
Version 3 |
| An ordinal $\alpha$ is called \emph{additively indecomposable} if it is not $0$ and for any $\beta,\gamma<\alpha$, $\beta+\gamma<\alpha$. |
An ordinal $\alpha$ is called \emph{additively indecomposable} if it is not $0$ and for any $\beta,\gamma<\alpha$, $\beta+\gamma<\alpha$. The set of additively indecomposable ordinals is denoted $\mathbb{H}$. |
| The set of additively indecomposable ordinals is denoted $\H$. |
Obviously $1\in\mathbb{H}$, since $0+0<1$. |
| Obviously $1\in\H$, since $0+0<1$. |
No finite ordinals other than $1$ are in $\mathbb{H}$. |
| No finite ordinal other than $1$ is in $\H$. |
Also, $\omega\in\mathbb{H}$ since the sum of two finite ordinals is still finite, |
| Also, $\omega\in\H$, since the sum of two finite ordinals is still finite. |
$\mathbb{H}$ is closed and unbounded, so the enumerating function of $\mathbb{H}$ is normal. In fact, $f_\mathbb{H}(\alpha)=\omega^\alpha$. |
| More generally, every infinite cardinal is in $\H$. |
The derivative $f_\mathbb{H}^\prime(\alpha)$ is written $\epsilon_\alpha$. The number $\epsilon_0=\omega^{\omega^{\omega^{\cdot^{\cdot^\cdot}}}}$ is therefore the first fixed point of the series |
| $\H$ is closed and unbounded, so the enumerating function of $\H$ is normal. |
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| In fact, $f_\H(\alpha)=\omega^\alpha$. |
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| The derivative $f_\H^\prime(\alpha)$ is written $\epsilon_\alpha$. |
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| The number $\epsilon_0=\omega^{\omega^{\omega^{\cdot^{\cdot^\cdot}}}}$ is therefore the first fixed point of the series |
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| $\omega,\omega^\omega\!,\omega^{\omega^\omega}\!\!,\ldots$ |
$\omega,\omega^\omega\!,\omega^{\omega^\omega}\!\!,\ldots$ |
