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 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 "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.


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
title=New Computational Paradigms
DOI 10.1007/b136981
ISBN =978-3-540-26179-7
DOI =10.1007/11494645_26
pages =211-220

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

*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=|title=Predicativity beyond Gamma_0|year=2005

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