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), ψα(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 or or . 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, JSTOR 1988605
- Weaver, Nik (2005), Predicativity beyond Gamma_0, arXiv:math/0509244, Bibcode:2005math......9244W
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- First infinite ordinal ω
- Epsilon numbers ε0
- Feferman–Schütte ordinal Γ0
- Ackermann ordinal θ(Ω2)
- small Veblen ordinal θ(Ωω)
- large Veblen ordinal θ(ΩΩ)
- Bachmann–Howard ordinal ψ(εΩ+1)
- Buchholz's ordinal ψ0(Ωω)
- Takeuti–Feferman–Buchholz ordinal ψ(εΩω+1)
- Proof-theoretic ordinals of the theories of iterated inductive definitions
- Computable ordinals < ωCK
1 - Nonrecursive ordinal ≥ ωCK
1 - First uncountable ordinal Ω
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