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.

There is no standard notation for ordinals beyond the Feferman–Schütte ordinal $\Gamma_0$ . Most systems of notation use symbols such as $\psi(\alpha)$ , $\theta(\alpha)$ , $\psi_\alpha(\beta)$ , 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 $\theta_{\Omega^\omega}(0)$ or $\psi(\Omega^{\Omega^\omega})$ 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 {{harv|Jervell|2005}}.

References

{{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}}

Category:Ordinal numbers