{"history":[],"article":{"id":"11a9eadd-5933-42f8-959d-3dadf343f8e2","title":"Additively indecomposable ordinal","slug":"additively-indecomposable-ordinal","content":":::summary\n:::\n\n# \nIn [set theory](https://bliptext.com/articles/set-theory), a branch of [mathematics](https://bliptext.com/articles/mathematics), an **additively indecomposable ordinal** α is any [ordinal number](https://bliptext.com/articles/ordinal-number) that is not 0 such that for any, we have Additively indecomposable ordinals were named the gamma numbers by Cantor, p.20 and are also called additive principal numbers.\nThe [class](https://bliptext.com/articles/class-set-theory) of additively indecomposable ordinals may be denoted \\mathbb H, from the German \"Hauptzahl\".\nThe additively indecomposable ordinals are precisely those ordinals of the form for some ordinal \\beta.\nFrom the continuity of addition in its right argument, we get that if and α is additively indecomposable, then\nObviously 1 is additively indecomposable, since 0+0<1.\nNo [finite](https://bliptext.com/articles/finite-set) ordinal other than 1 is additively indecomposable.\nAlso, \\omega is additively indecomposable, since the sum of two finite ordinals is still finite.\nMore generally, every [infinite](https://bliptext.com/articles/infinite-set) [initial ordinal](https://bliptext.com/articles/initial-ordinal) (an ordinal corresponding to a [cardinal number](https://bliptext.com/articles/cardinal-number)) is additively indecomposable.\nThe class of additively indecomposable numbers is [closed](https://bliptext.com/articles/order-topology) and unbounded.\nIts enumerating function is [normal](https://bliptext.com/articles/normal-function), given by.\nThe derivative of (which enumerates its [fixed points](https://bliptext.com/articles/fixed-point-mathematics)) is written Ordinals of this form (that is, [fixed points](https://bliptext.com/articles/fixed-point-mathematics) of ) are called [epsilon numbers](https://bliptext.com/articles/epsilon-numbers-mathematics).\nThe number is therefore the first fixed point of the [sequence](https://bliptext.com/articles/sequence)\n\n# Multiplicatively indecomposable\nA similar notion can be defined for multiplication.\nIf α is greater than the multiplicative identity, 1, and β < α and γ < α imply β·γ < α, then α is multiplicatively indecomposable.\nThe finite ordinal 2 is multiplicatively indecomposable since 1·1 = 1 < 2.\nBesides 2, the multiplicatively indecomposable ordinals (named the delta numbers by Cantor p.20) are those of the form for any ordinal α.\nEvery [epsilon number](https://bliptext.com/articles/epsilon-numbers-mathematics) is multiplicatively indecomposable; and every multiplicatively indecomposable ordinal (other than 2) is additively indecomposable.\nThe delta numbers (other than 2) are the same as the [prime ordinals](https://bliptext.com/articles/prime-ordinal) that are limits.\n\n# Higher indecomposables\nExponentially indecomposable ordinals are equal to the epsilon numbers, tetrationally indecomposable ordinals are equal to the zeta numbers (fixed points of ), and so on.\nTherefore, is the first ordinal which is \\uparrow^n-indecomposable for all n, where \\uparrow denotes [Knuth's up-arrow notation](https://bliptext.com/articles/knuth-s-up-arrow-notation).","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T10:37:13.208Z","updated_at":"2025-01-26T10:37:13.208Z","search_vector":"'addit':1 'indecompos':2 'ordin':3","search_content":" \n \n\n# \nIn [set theory](https://bliptext.com/articles/set-theory), a branch of [mathematics](https://bliptext.com/articles/mathematics), an **additively indecomposable ordinal** α is any [ordinal number](https://bliptext.com/articles/ordinal-number) that is not 0 such that for any, we have Additively indecomposable ordinals were named the gamma numbers by Cantor, p.20 and are also called additive principal numbers.\nThe [class](https://bliptext.com/articles/class-set-theory) of additively indecomposable ordinals may be denoted \\mathbb H, from the German \"Hauptzahl\".\nThe additively indecomposable ordinals are precisely those ordinals of the form for some ordinal \\beta.\nFrom the continuity of addition in its right argument, we get that if and α is additively indecomposable, then\nObviously 1 is additively indecomposable, since 0+0<1.\nNo [finite](https://bliptext.com/articles/finite-set) ordinal other than 1 is additively indecomposable.\nAlso, \\omega is additively indecomposable, since the sum of two finite ordinals is still finite.\nMore generally, every [infinite](https://bliptext.com/articles/infinite-set) [initial ordinal](https://bliptext.com/articles/initial-ordinal) (an ordinal corresponding to a [cardinal number](https://bliptext.com/articles/cardinal-number)) is additively indecomposable.\nThe class of additively indecomposable numbers is [closed](https://bliptext.com/articles/order-topology) and unbounded.\nIts enumerating function is [normal](https://bliptext.com/articles/normal-function), given by.\nThe derivative of (which enumerates its [fixed points](https://bliptext.com/articles/fixed-point-mathematics)) is written Ordinals of this form (that is, [fixed points](https://bliptext.com/articles/fixed-point-mathematics) of ) are called [epsilon numbers](https://bliptext.com/articles/epsilon-numbers-mathematics).\nThe number is therefore the first fixed point of the [sequence](https://bliptext.com/articles/sequence)\n\n# Multiplicatively indecomposable\nA similar notion can be defined for multiplication.\nIf α is greater than the multiplicative identity, 1, and β < α and γ < α imply β·γ < α, then α is multiplicatively indecomposable.\nThe finite ordinal 2 is multiplicatively indecomposable since 1·1 = 1 < 2.\nBesides 2, the multiplicatively indecomposable ordinals (named the delta numbers by Cantor p.20) are those of the form for any ordinal α.\nEvery [epsilon number](https://bliptext.com/articles/epsilon-numbers-mathematics) is multiplicatively indecomposable; and every multiplicatively indecomposable ordinal (other than 2) is additively indecomposable.\nThe delta numbers (other than 2) are the same as the [prime ordinals](https://bliptext.com/articles/prime-ordinal) that are limits.\n\n# Higher indecomposables\nExponentially indecomposable ordinals are equal to the epsilon numbers, tetrationally indecomposable ordinals are equal to the zeta numbers (fixed points of ), and so on.\nTherefore, is the first ordinal which is \\uparrow^n-indecomposable for all n, where \\uparrow denotes [Knuth's up-arrow notation](https://bliptext.com/articles/knuth-s-up-arrow-notation).","revisionCount":0},"hasMore":false,"nextCursor":null}

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