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 harvtxt|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 $phi_$ Omega^omega(0) or $heta(Omega^omega)$ or $psi(Omega^omega)$ 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 harv|Jervel|2005.

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