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 Ackermann (1951) is somewhat smaller than the small Veblen 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), 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 small Veblen ordinal ${\displaystyle \phi _{\Omega ^{\omega }}(0)}$ or ${\displaystyle \theta (\Omega ^{\omega })}$ or ${\displaystyle \psi (\Omega ^{\Omega ^{\omega }})}$ 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 (Jervell 2005).