Small Veblen ordinal explained

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.

\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 functions to produce countable ordinals even for uncountable arguments, and some of which are "collapsing functions".

The small Veblen ordinal

\theta _{\Omega ^{\omega}} ( 0 ) or \psi(\Omega ^{\Omega ^{\omega}} ) is the limit of ordinals that can be described using a version of Veblen functions with finitely many arguments. It is the ordinal that measures the strength of Kruskal's theorem. It is also the ordinal type of a certain ordering of rooted trees .

References

• • • • • Weaver. Nik. math/0509244. Predicativity beyond Gamma_0. 2005 . cs2.