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...up casket]] & [[User:Fool|Fool]]|desc=This spiral represents all [[ordinal numbers]] less than <math>\omega^\omega</math>.|class={{{class}}}}} ...

| caption = spiral figure representing both finite and transfinite ordinal numbers ...ity with the [[cardinality of the continuum|cardinality of the set of real numbers]] can neither be proved nor disproved within the standard version of [[axio ...

...re precisely those ordinals of the form <math>\omega^\beta</math> for some ordinal <math>\beta</math>. ...e. More generally, every [[infinite set|infinite]] [[initial ordinal]] (an ordinal corresponding to a [[cardinal number]]) is additively indecomposable. ...

{{Short description|Smallest ordinal number that, considered as a set, is uncountable}} ...ath>\omega_1</math> or sometimes by <math>\Omega</math>, is the smallest [[ordinal number]] that, considered as a [[set (mathematics)|set]], is [[uncountable] ...

...der type of the segment bounded by <math>D_0D_\omega0</math> in Buchholz's ordinal notation <math>\mathsf{(OT, <)}</math>.<ref name=":0" /> Lastly, it can be * <math>\psi_i(\alpha)</math> is the smallest ordinal not in <math>C_i(\alpha)</math>. ...

...><ref>{{Cite journal|date=1986-01-01|title=A new system of proof-theoretic ordinal functions|url=|journal=Annals of Pure and Applied Logic|language=en|volume= ...ger |isbn=978-3-540-07533-2 |language=de}}</ref> It is the proof-theoretic ordinal of several formal theories: ...

...some authors may allow fundamental sequences to be defined on [[successor ordinal]]s.<ref name="BCW94">W. Buchholz, A. Cichon, A. Weiermann, [https://epub.ub Given an ordinal <math>\alpha</math>, a '''fundamental sequence''' for <math>\alpha</math> i ...

...hard|last2=Rado|author2-link=Richard Rado|year=1965}}, states that every [[ordinal number]] <math>\alpha</math> less than the [[Successor cardinal|successor]] ...roof is by transfinite induction. Let ''<math>\alpha</math>'' be a limit ordinal (the induction is trivial for successor ordinals), and for each ''<math>\be ...

...>:&nbsp;'''N'''&nbsp;→&nbsp;'''N''' (where '''N''' is the set of [[natural numbers]], {0,&nbsp;1,&nbsp;...}) called '''Hardy functions'''. It is related to th ...=G. H. Hardy |year=1904 |title=A THEOREM CONCERNING THE INFINITE CANONICAL NUMBERS |journal=Quarterly Journal of Mathematics |volume=35 |pages=87&ndash;94}}</ ...

...n be used to define [[representative (mathematics)|representative]]s for [[ordinal number]]s in ZF, [[Zermelo–Fraenkel set theory]] without the axiom of choic ...umbers, Scott's trick can be used to obtain representatives for [[cardinal numbers]] and more generally for [[Isomorphism class|isomorphism types]], for examp ...

...imitive recursive function]]s, defined for [[Set (mathematics)|sets]] or [[Ordinal number|ordinals]] rather than [[natural number]]s. They were introduced by ...'} (the [[Successor ordinal|successor]] of ''x''). The primitive recursive ordinal functions are the same as the primitive recursive set functions that map or ...

This is the [[Riesz representation theorem]] on the [[Ordinal (mathematics)|ordinal]] <math>\omega</math>. ...

...[normal function]]s closed in <math>M</math> are closed under some regular ordinal <math>< M</math>). Rathjen uses this to diagonalise over the [[Inaccessible .../math>, which is called the "Small Rathjen ordinal" is the proof-theoretic ordinal of <math>\mathsf{KPM}</math>, Kripke–Platek set theory augmented by the axi ...

To define the system of ν-times iterated inductive definitions, where ν is an ordinal, let <math>\prec</math> be a primitive recursive well-ordering of order typ * The proof-theoretic ordinal of ID_<ν is equal to <math>\psi_0(\Omega_\nu)</math>. ...

...ect in successive equal time intervals is linearly proportional to the odd numbers. That is, if a body falling from rest covers a certain distance during an a ...eo's_law_of_odd_numbers.svg|thumb|300px|Derivation of Galileo's law of odd numbers]] ...

...'g''_α: '''N''' → '''N''' (where '''N''' is the set of [[natural numbers]], {0, 1, ...}). It contrasts with the [[fast-growing hierarchy]]. ...nn.edu/~jean/kruskal.pdf What's so special about Kruskal's theorem and the ordinal Γ₀? A survey of some results in proof theory] (2012, p.63). Acce ...

...notion of height admits a refinement so that the ''p''-height becomes an [[ordinal number]]. Height plays an important role in [[Prüfer theorems]] and also in For any ordinal ''α'', there is a subgroup ''p''^''α''''A'' of ''A'' which is the ...

{{Short description|Ordinal-indexed family of rapidly increasing functions: ℕ→ℕ}} ...or '''Löb–Wainer hierarchy''', which is an extension to all α < [[Epsilon numbers (mathematics)|ε₀]]. Such hierarchies provide a natural way to c ...

...z's psi-functions''' are a hierarchy of single-argument [[Ordinal notation|ordinal functions]] <math>\psi_\nu(\alpha)</math> introduced by German mathematicia ...functions &psi;_''v''(&alpha;) for &alpha; an ordinal, ''v'' an ordinal at most &omega;, are defined by induction on &alpha; as follows: ...

...is the set of possible outcomes. E.g., it may be the set of positive real numbers, representing the possible annual [[gross domestic product]]. It is normali == Ordinal definition == ...