Large Veblen ordinal

In mathematics, the large Veblen ordinal is a certain large countable ordinal, named after Oswald Veblen.

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 ordinal collapsing functions.

The large Veblen ordinal is sometimes denoted by $\phi _{{{\Omega ^{\Omega }}}}(0)$ or $\theta (\Omega ^{\Omega })$ or $\psi (\Omega ^{{\Omega ^{\Omega }}})$ . It was constructed by Veblen using an extension of Veblen functions allowing infinitely many arguments.

References

Veblen, Oswald (1908), "Continuous Increasing Functions of Finite and Transfinite Ordinals", Transactions of the American Mathematical Society, 9 (3): 280–292, doi:10.2307/1988605
Weaver, Nik (2005), Predicativity beyond Gamma_0, arXiv:math/0509244

Large countable ordinals

First infinite ordinal ω Epsilon numbers ε₀ Feferman–Schütte ordinal Γ₀ Ackermann ordinal θ(Ω²) small Veblen ordinal θ(Ω^ω) large Veblen ordinal θ(Ω^Ω) Bachmann–Howard ordinal ψ(ε_Ω+1) Ψ₀(Ω_ω) Church–Kleene ordinal ω₁^CK

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