Ackermann ordinal

Short description: Certain large countable ordinal

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.

Unfortunately there is no standard notation for ordinals beyond the Feferman–Schütte ordinal Γ0. Most systems of notation use symbols such as ψ(α), θ(α), ψα(β), some of which are modifications of the Veblen functions to produce countable ordinals even for uncountable arguments, and some of which are "collapsing functions". 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 (Ackermann 1951), and is sometimes denoted by [math]\displaystyle{ \varphi_{{\Omega^2}}(0) }[/math] or [math]\displaystyle{ \theta(\Omega^2) }[/math], [math]\displaystyle{ \psi(\Omega^{\Omega^2}) }[/math], or [math]\displaystyle{ \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 (Veblen 1908), which he seems to have been unaware of.

## References

* Ackermann, Wilhelm (1951), "Konstruktiver Aufbau eines Abschnitts der zweiten Cantorschen Zahlenklasse", Math. Z. 53 (5): 403–413, doi:10.1007/BF01175640
* Veblen, Oswald (1908), "Continuous Increasing Functions of Finite and Transfinite Ordinals", Transactions of the American Mathematical Society 9 (3): 280–292, doi:10.2307/1988605
* Weaver, Nik (2005), "Predicativity beyond Γ0", arXiv:math/0509244

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Large countable ordinals

* First infinite ordinal ω
* Epsilon numbers ε₀
* Feferman–Schütte ordinal Γ0
* Ackermann ordinal θ(Ω3)
* small Veblen ordinal θ(Ωω)
* large Veblen ordinal θ(ΩΩ)
* Bachmann–Howard ordinal ψ(εΩ+1)
* Ψ0(Ωω)
* Church–Kleene ordinal ω1CK

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