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additively indecomposable (Definition)

An ordinal $ \alpha$ is called additively indecomposable if it is not 0 and for any $ \beta,\gamma<\alpha$, we have $ \beta+\gamma<\alpha$. The set of additively indecomposable ordinals is denoted $ \mathbb{H}$.

Obviously $ 1\in\mathbb{H}$, since $ 0+0<1$. No finite ordinal other than $ 1$ is in $ \mathbb{H}$. Also, $ \omega\in\mathbb{H}$, since the sum of two finite ordinals is still finite. More generally, every infinite cardinal is in $ \mathbb{H}$.

$ \mathbb{H}$ is closed and unbounded, so the enumerating function of $ \mathbb{H}$ is normal. In fact, $ f_\mathbb{H}(\alpha)=\omega^\alpha$.

The derivative $ f_\mathbb{H}^\prime(\alpha)$ is written $ \epsilon_\alpha$. Ordinals of this form (that is, fixed points of $ f_\mathbb{H}$) are called epsilon numbers. The number $ \epsilon_0=\omega^{\omega^{\omega^{\cdot^{\cdot^\cdot}}}}$ is therefore the first fixed point of the series $ \omega,\omega^\omega\!,\omega^{\omega^\omega}\!\!,\ldots$



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"additively indecomposable" is owned by mathcam. [ full author list (3) | owner history (2) ]
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See Also: ordinal arithmetic

Also defines:  epsilon number, epsilon zero

Attachments:
proof of theorems in additively indecomposable (Proof) by mathcam
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Cross-references: series, number, fixed points, derivative, normal, enumerating function, unbounded, closed, cardinal, infinite, sum, finite, ordinal
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This is version 8 of additively indecomposable, born on 2003-02-23, modified 2005-03-03.
Object id is 4056, canonical name is AdditivelyIndecomposable.
Accessed 5264 times total.

Classification:
AMS MSC: 03E10 (Mathematical logic and foundations :: Set theory :: Ordinal and cardinal numbers)
 03F15 (Mathematical logic and foundations :: Proof theory and constructive mathematics :: Recursive ordinals and ordinal notations)
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