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abstract
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In set theory, a branch of mathematics, an additively indecomposable ordinal α 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> Obviously <math>1\in\mathbb{H}</math>, since <math>0+0<1. </math> No 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 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> is written <math>\epsilon_\alpha. </math> Ordinals of this form (that is, fixed points of <math>f_\mathbb{H}</math>) are called 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>
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4056
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title
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Additively indecomposable
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wiki page uses template
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http://dbpedia.org/resource/Template:planetmath
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comment
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In set theory, a branch of mathematics, an additively indecomposable ordinal α 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> Obviously <math>1\in\mathbb{H}</math>, since <math>0+0<1.
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label
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Additively indecomposable ordinal
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sameAs
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http://rdf.freebase.com/ns/guid.9202a8c04000641f80000000083bd076
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subject
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http://dbpedia.org/resource/Category:Ordinal_numbers
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sourceURL
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Additively indecomposable ordinal
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page
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http://en.wikipedia.org/wiki/Additively_indecomposable_ordinal
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http://dbpedia.org/resource/Additively_indecomposable
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