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.

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 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 , and is sometimes denoted by $\varphi_(0)$ or $\theta(\Omega^2)$, $\psi(\Omega^)$, or $\varphi(1,0,0,0)$, where Ω is the smallest uncountable ordinal. Ackermann's system of notation is weaker than the system introduced much earlier by , which he seems to have been unaware of.

References

*
*
*

Ordinal numbers

{{number-stub