Automatic calculation of particle interaction or decay: Difference between revisions

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In [[set theory]], a branch of [[mathematics]], an '''additively indecomposable ordinal''' &alpha; is any [[ordinal number]] that is not 0 such that for any  <math>\beta,\gamma<\alpha</math>, we have  <math>\beta+\gamma<\alpha.</math> The set of additively indecomposable ordinals is denoted  <math>\mathbb{H}.</math>
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From the continuity of addition in its right argument, we get that if <math>\beta < \alpha</math> and &alpha; is additively indecomposable, then <math>\beta + \alpha = \alpha.</math>
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Obviously  <math>1\in\mathbb{H}</math>, since <math>0+0<1.</math> No [[finite set|finite]] ordinal other than <math>1</math> is in <math>\mathbb{H}.</math> Also, <math>\omega\in\mathbb{H}</math>, since the sum of two finite ordinals is still finite. More generally, every [[infinite set|infinite]] [[cardinal number|cardinal]] is in <math>\mathbb{H}.</math>
 
<math>\mathbb{H}</math> is closed and unbounded, so the enumerating function of <math>\mathbb{H}</math> is normal. In fact, <math>f_\mathbb{H}(\alpha)=\omega^\alpha.</math>
 
The derivative  <math>f_\mathbb{H}^\prime(\alpha)</math> (which enumerates fixed points of ''f''<sub>H</sub>) is written  <math>\epsilon_\alpha.</math> Ordinals of this form (that is, [[fixed point (mathematics)|fixed point]]s of  <math>f_\mathbb{H}</math>) are called ''[[epsilon nought|epsilon numbers]]''. The number <math>\epsilon_0=\omega^{\omega^{\omega^{\cdot^{\cdot^\cdot}}}}</math> is therefore the first fixed point of the [[sequence]]  <math>\omega,\omega^\omega\!,\omega^{\omega^\omega}\!\!,\ldots</math>
 
== Multiplicatively indecomposable ==
A similar notion can be defined for multiplication. The multiplicatively indecomposable ordinals are those of the form <math>\omega^{\omega^\alpha} \,</math> for any ordinal &alpha;.
 
== See also ==
* [[Ordinal arithmetic]]
 
{{PlanetMath attribution|id=4056|title=Additively indecomposable}}
 
[[Category:Ordinal numbers]]

Latest revision as of 13:24, 5 May 2014

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