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{{short description|Certain large countable ordinal}}
{{About|a countable ordinal||Ackermann (disambiguation)}}
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. 

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 last one is an extension of the Veblen functions for more than 2 arguments.

The smaller Ackermann ordinal is the limit of a system of ordinal notations invented by {{harvtxt|Ackermann|1951}}, and is sometimes denoted by <math>\varphi_{{\Omega^2}}(0)</math> or  <math>\theta(\Omega^2)</math>, <math>\psi(\Omega^{\Omega^2})</math>, or <math>\varphi(1,0,0,0)</math>, where Ω is the [[smallest uncountable ordinal]]. 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==
{{refbegin}}
*{{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|s2cid=119687180}}
*{{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|jstor=1988605 |doi-access=free}}
*{{cite arXiv |last=Weaver|first=Nik|eprint=math/0509244|title=Predicativity beyond &Gamma;<sub>0</sub>|year=2005|mode=cs2}}
{{refend}}

{{countable ordinals}}

{{DEFAULTSORT:Ackermann Ordinal}}
[[Category:Ordinal numbers]]


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