# Small Veblen ordinal

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 &psi;(α), &theta;(α), &psi;α(&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 $phi_$Omega^omega(0) or $heta\left(Omega^omega\right)$ or $psi\left(Omega^omega\right)$ 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.

References

*citation|id=MR|0039669
last=Ackermann|first= Wilhelm
title=Konstruktiver Aufbau eines Abschnitts der zweiten Cantorschen Zahlenklasse
journal=Math. Z. |volume=53|year=1951|pages= 403-413|doi= 10.1007/BF01175640

*citation|chapter= Finite Trees as Ordinals
first=Herman Ruge |last=Jervel
series= Lecture Notes in Computer Science
publisher =Springer |place=Berlin / Heidelberg
ISSN = 1611-3349
volume =3526
DOI 10.1007/b136981
year=2005
ISBN =978-3-540-26179-7
DOI =10.1007/11494645_26
pages =211-220

*citation|id=MR|1212407
last=Rathjen|first= Michael|last2= Weiermann|first2= Andreas
title=Proof-theoretic investigations on Kruskal's theorem
journal=Ann. Pure Appl. Logic|volume= 60 |year=1993|issue= 1|pages= 49-88
doi=10.1016/0168-0072(93)90192-G

*citation|title= Continuous Increasing Functions of Finite and Transfinite Ordinals
first= Oswald |last=Veblen
journal= Transactions of the American Mathematical Society|volume= 9|issue= 3|year= 1908|pages=280-292

*citation|last=Weaver|first=Nik|url=http://arxiv.org/abs/math/0509244|title=Predicativity beyond Gamma_0|year=2005

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