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.

There is no standard notation for ordinals beyond the Feferman–Schütte ordinal ${\displaystyle \Gamma _{0}}$. Most systems of notation use symbols such as ${\displaystyle \psi (\alpha )}$, ${\displaystyle \theta (\alpha )}$, ${\displaystyle \psi _{\alpha }(\beta )}$, 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 ${\displaystyle \theta _{\Omega ^{\omega }}(0)}$ 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).

References

• Ackermann, Wilhelm (1951), "Konstruktiver Aufbau eines Abschnitts der zweiten Cantorschen Zahlenklasse", Math. Z., 53 (5): 403–413, doi:10.1007/BF01175640, MR 0039669, S2CID 119687180
• Jervell, Herman Ruge (2005), "Finite Trees as Ordinals" (PDF), New Computational Paradigms, Lecture Notes in Computer Science, vol. 3526, Berlin / Heidelberg: Springer, pp. 211–220, doi:10.1007/11494645_26, ISBN 978-3-540-26179-7
• Rathjen, Michael; Weiermann, Andreas (1993), "Proof-theoretic investigations on Kruskal's theorem", Ann. Pure Appl. Logic, 60 (1): 49–88, doi:10.1016/0168-0072(93)90192-G, MR 1212407
• 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