In mathematics, the small Veblen 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

Oswald Veblen Oswald Veblen (June 24, 1880 – August 10, 1960) was an American mathematician, geometer and topologist, whose work found application in atomic physics and the theory of relativity. He proved the Jordan curve theorem in 1905; while this was lon ...

. It is occasionally called the Ackermann ordinal, though the

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 ...

described by is somewhat smaller than the small Veblen ordinal. 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, ...

\Gamma_0

. Most systems of notation use symbols such as

\psi(\alpha)

\theta(\alpha)

\psi_\alpha(\beta)

, 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 function In mathematical logic and set theory, an ordinal collapsing function (or projection function) is a technique for defining (notations for) certain recursive large countable ordinals, whose principle is to give names to certain ordinals much larger ...

s". The small Veblen ordinal

\theta_(0)

\psi(\Omega^)

is the limit of ordinals that can be described using a version of

s with finitely many arguments. It is the ordinal that measures the strength of

Kruskal's theorem In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding. History The theorem was conjectured by Andrew Vázsonyi and proved by ; ...

. It is also the ordinal type of a certain ordering of

rooted tree In graph theory, a tree is an undirected graph in which any two vertices are connected by ''exactly one'' path, or equivalently a connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by ''a ...

s .

References

* * * * * {{countable ordinals Ordinal numbers