Ackermann 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), ψ_α(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 smaller Ackermann ordinal is the limit of a system of ordinal notations invented by Ackermann (1951), and is sometimes denoted by $\phi _{{{\Omega ^{2}}}}(0)$ or $\theta (\Omega ^{2})$ or $\psi (\Omega ^{{\Omega ^{2}}})$ . 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, MR 0039669
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 Gamma_0, arXiv:math/0509244

Large countable ordinals

First infinite ordinal ω Epsilon numbers ε₀ Feferman–Schütte ordinal Γ₀ Ackermann ordinal θ(Ω²) small Veblen ordinal θ(Ω^ω) large Veblen ordinal θ(Ω^Ω) Bachmann–Howard ordinal ψ(ε_Ω+1) Ψ₀(Ω_ω) Church–Kleene ordinal ω₁^CK

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