In mathematics, the Ackermann ordinal is a certain
large countable ordinal
In the mathematical discipline of set theory, there are many ways of describing specific countable ordinals. The smallest ones can be usefully and non-circularly expressed in terms of their Cantor normal forms. Beyond that, many ordinals of rele ...
, named after
Wilhelm Ackermann
Wilhelm Friedrich Ackermann (; ; 29 March 1896 – 24 December 1962) was a German mathematician and logician best known for his work in mathematical logic and the Ackermann function, an important example in the theory of computation.
Biography
...
. The term "Ackermann ordinal" is also occasionally used for the
small Veblen ordinal
In mathematics, the small Veblen ordinal is a certain large countable ordinal, named after Oswald Veblen. It is occasionally called the Ackermann ordinal, though the Ackermann ordinal described by is somewhat smaller than the small Veblen ordina ...
, a somewhat larger ordinal.
Unfortunately there is no standard notation for ordinals beyond the
Feferman–Schütte ordinal
In mathematics, the Feferman–Schütte ordinal Γ0 is a large countable ordinal.
It is the proof-theoretic ordinal of several mathematical theories, such as arithmetical transfinite recursion.
It is named after Solomon Feferman and Kurt Schütte, ...
Γ
0. Most systems of notation use symbols such as ψ(α), θ(α), ψ
α(β), some of which are modifications of the
Veblen function
In mathematics, the Veblen functions are a hierarchy of normal functions ( continuous strictly increasing functions from ordinals to ordinals), introduced by Oswald Veblen in . If φ0 is any normal function, then for any non-zero ordinal α, φ ...
s 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
or
,
, or
, 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
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Ordinal numbers
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