View source for Small Veblen ordinal

{{short description|Certain large countable 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.

There is no standard notation for ordinals beyond the [[Feferman–Schütte ordinal]] <math>\Gamma_0</math>. Most systems of notation use symbols such as <math>\psi(\alpha)</math>, <math>\theta(\alpha)</math>, <math>\psi_\alpha(\beta)</math>, some of which are modifications of the [[Veblen function]]s to produce countable ordinals even for uncountable arguments, and some of which are "[[collapsing function]]s".

The small Veblen ordinal <math>\theta_{\Omega^\omega}(0)</math> or <math>\psi(\Omega^{\Omega^\omega})</math> is the limit of ordinals that can be described using a version of [[Veblen function]]s 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 tree]]s {{harv|Jervell|2005}}.
<!--This is the supremum of order types of a class of [[Rewrite_order#Formal_definitions|rewrite orderings]] called "simplification orderings".<ref>H. Levitz, [https://www.ccs.neu.edu/home/pete/courses/Formal-methods/2008-Fall/readings/transfinite-ordinals-notations.pdf Transfinite Ordinals and Their Notations: For The Uninitiated] (2008, p.8). Accessed 13 September 2022.</ref>-->

==References==
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*{{citation|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|issue=5|s2cid=119687180}}
*{{citation |chapter-url=http://folk.uio.no/herman/finord.pdf |first=Herman Ruge |last=Jervell |series=Lecture Notes in Computer Science |publisher=Springer |place=Berlin / Heidelberg |volume=3526 |title=New Computational Paradigms |year=2005 |isbn=978-3-540-26179-7 |doi=10.1007/11494645_26 |pages=[https://archive.org/details/newcomputational0000conf/page/211 211–220] |chapter=Finite Trees as Ordinals |url=https://archive.org/details/newcomputational0000conf/page/211 }}
*{{citation|mr=1212407 |last1=Rathjen|first1= 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 |url=http://www.amsta.leeds.ac.uk/Pure/staff/rathjen/kruskal.ps |doi=10.1016/0168-0072(93)90192-G|doi-access=free}} 
*{{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 |jstor=1988605|doi= 10.2307/1988605|doi-access=free}}
*{{cite arXiv |last=Weaver|first=Nik|eprint=math/0509244|title=Predicativity beyond Gamma_0|year=2005 |mode=cs2}}
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{{countable ordinals}}

[[Category:Ordinal numbers]]


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