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In mathematics, the '''Ackermann ordinal''' is a certain [[large countable ordinal]], named after [[Wilhelm Ackermann]]. The term "Ackermann ordinal" is also occasionally used for the [[small Veblen ordinal]], a somewhat larger ordinal.  
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Unfortunately there is no standard notation for ordinals beyond the [[Feferman–Schütte ordinal]] Γ<sub>0</sub>. Most systems of notation use symbols such as ψ(α), θ(α), ψ<sub>α</sub>(β), some of which are modifications of the [[Veblen function]]s to produce countable ordinals even for uncountable arguments, and some of which are "[[collapsing function]]s".
 
The smaller '''Ackermann ordinal''' is the limit of a system of ordinal notations invented by {{harvtxt|Ackermann|1951}}, and is sometimes denoted by <math>\phi_{{\Omega^2}}(0)</math> or  <math>\theta(\Omega^2)</math> or <math>\psi(\Omega^{\Omega^2})</math>.  Ackermann's system of notation is weaker than the system introduced much earlier by {{harvtxt|Veblen|1908}}, which he seems to have been unaware of.
 
==References==
*{{citation|mr=0039669|last=Ackermann|first= Wilhelm |title=Konstruktiver Aufbau eines Abschnitts der zweiten Cantorschen Zahlenklasse
|journal=Math. Z. |volume=53|year=1951|pages= 403–413|doi=10.1007/BF01175640|issue=5}}
*{{citation |title= Continuous Increasing Functions of Finite and Transfinite Ordinals |first=Oswald |last=Veblen
|journal=Transactions of the American Mathematical Society |volume= 9|issue= 3|year= 1908|pages=280–292 |doi= 10.2307/1988605}}
*{{citation|last=Weaver|first=Nik|arxiv=math/0509244|title=Predicativity beyond Gamma_0|year=2005}}
{{countable ordinals}}
 
{{DEFAULTSORT:Ackermann Ordinal}}
[[Category:Ordinal numbers]]

Latest revision as of 08:44, 14 November 2014

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