Veblen function: Difference between revisions

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For example, if <u>α</u>=(1@&omega;) denotes the transfinite sequence with value 1 at ω and 0 everywhere else, then φ(1@ω) is the smallest fixed point of all the functions ξ↦φ(ξ,0,...,0) with finitely many final zeroes (it is also the limit of the φ(1,0,...,0) with finitely many zeroes, the small Veblen ordinal).

The smallest ordinal α such that α is greater than φ applied to any function with support in α (i.e., which cannot be reached "from below" using the Veblen function of transfinitely many variables) is sometimes known as the [[Large Veblen ordinal|"large" Veblen ordinal]], or "great" Veblen number.<ref>M. Rathjen, "[https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf The Art of Ordinal Analysis]" (2006), appearing in Proceedings of the International Congress of Mathematicians 2006.</ref>.

==Values==

The function takes on several prominent values:

* <math>\varphi(\omega,0)</math>, a bound on the order types of the [[Path ordering (term rewriting)|recursive path orderings]] with finitely many function symbols. <ref>M. Dershowitz, N. Okada, [https://www.cs.tau.ac.il/~nachumd/papers/ProofTheoretic.pdf Proof Theoretic Techniques for Term Rewriting Theory] (1988). p.105</ref>

* The [[Feferman-Schutte ordinal]] <math>\Gamma_0</math> is equal to <math>\varphi(1,0,0)</math>.<ref>D. Madore, "[http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals]" (2017). Accessed 02 November 2022.</ref>

* The [[small Veblen ordinal]] is equal to <math>\varphi\begin{pmatrix}1 \\ \omega\end{pmatrix}</math>. <ref>F. Ranzi, T. Strahm, "[https://link.springer.com/content/pdf/10.1007/s00153-019-00658-x.pdf A flexible type system for the small Veblen ordinal] (p.730). Accessed 03 November 2022.</ref>

==References==

{{Reflist}}

* <!--Not considered reliable, according to a revert on [[Small Veblen ordinal]].-->Hilbert Levitz, ''[http://www.cs.fsu.edu/~levitz/ords.ps Transfinite Ordinals and Their Notations: For The Uninitiated]'', expository article (8 pages, in [[PostScript]])

*{{citation|last= Pohlers|first= Wolfram|title= Proof theory|mr= 1026933|series= Lecture Notes in Mathematics|volume= 1407|publisher= Springer-Verlag|place= Berlin|year= 1989|isbn= 978-3-540-51842-6|doi= 10.1007/978-3-540-46825-7|url-access= registration|url= https://archive.org/details/prooftheoryintro0000pohl}}

*{{citation|mr=0505313|last= Schütte|first= Kurt |title=Proof theory|series= Grundlehren der Mathematischen Wissenschaften|volume= 225|publisher= Springer-Verlag|place= Berlin-New York|year= 1977|pages= xii+299 | isbn= 978-3-540-07911-8}}