Ackermann ordinal
Encyclopedia
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 function
s 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 , and is sometimes denoted by
or 
or 
. Ackermann's system of notation is weaker than the system introduced much earlier by , which he seems to have been unaware of.
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 relevance to proof theory still have computable ordinal...
, named after Wilhelm Ackermann
Wilhelm Ackermann
Wilhelm Friedrich Ackermann was a German mathematician best known for the Ackermann function, an important example in the theory of computation....
. The term "Ackermann ordinal" is also occasionally used for the small Veblen ordinal
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 ordinal....
, a somewhat larger ordinal.
Unfortunately there is no standard notation for ordinals beyond the Feferman–Schütte ordinal
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....
Γ0. Most systems of notation use symbols such as ψ(α), θ(α), ψα(β), some of which are modifications of the Veblen function
Veblen function
In mathematics, the Veblen functions are a hierarchy of normal functions , introduced by Oswald Veblen in...
s 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 , and is sometimes denoted by
or or . Ackermann's system of notation is weaker than the system introduced much earlier by , which he seems to have been unaware of.