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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>
=== Further extensions ===
In {{harvtxt|Massmann|Kwon|2023}}, the Veblen function was extended further to a somewhat technical system known as ''dimensional Veblen''. In this, one may take fixed points or row numbers, meaning expressions such as φ(1@(1,0)) are valid (representing the large Veblen ordinal), visualised as multi-dimensional arrays. It was proven that all ordinals below the [[Bachmann–Howard ordinal]] could be represented in this system, and that the representations for all ordinals below the [[large Veblen ordinal]] were aesthetically the same as in the original system.
==Values==
The function takes on several prominent values:
* <math>\varphi(1,0) = \varepsilon_0</math> is the [[Proof theoretic ordinal|proof-theoretic ordinal]] [[Gentzen's consistency proof|of]] [[Peano axioms|Peano arithmetic]] and the limit of what ordinals can be represented in terms of [[Cantor normal form]] and smaller ordinals.
* <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, and the smallest ordinal closed under [[Primitive recursive function|primitive recursive]] ordinal functions.<ref>N. Dershowitz, M. Okada, [https://www.cs.tau.ac.il/~nachumd/papers/ProofTheoretic.pdf Proof Theoretic Techniques for Term Rewriting Theory] (1988). p.105</ref><ref>{{Cite journal |last=Avigad |first=Jeremy |date=May 23, 2001 |title=An ordinal analysis of admissible set theory using recursion on ordinal notations |url=https://www.andrew.cmu.edu/user/avigad/Papers/admissible.pdf |journal=Journal of Mathematical Logic |volume=2 |pages=91--112 |doi=10.1142/s0219061302000126}}</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>{{cite journal | url=https://link.springer.com/content/pdf/10.1007/s00153-019-00658-x.pdf | doi=10.1007/s00153-019-00658-x | title=A flexible type system for the small Veblen ordinal | year=2019 | last1=Ranzi | first1=Florian | last2=Strahm | first2=Thomas | journal=Archive for Mathematical Logic | volume=58 | issue=5–6 | pages=711–751 | s2cid=253675808 }}</ref>
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*{{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 |doi= 10.2307/1988605|jstor=1988605|doi-access= free}}
*{{citation |jstor=2272243 |pages=439–459 |last1=Miller |first1=Larry W. |title=Normal Functions and Constructive Ordinal Notations |volume=41 |issue=2 |journal=The Journal of Symbolic Logic |year=1976 |doi=10.2307/2272243}}
*{{citation |last=Massmann |first=Jayde Sylvie |title=Extending the Veblen Function |date=October 20, 2023 |url=https://arxiv.org/abs/2310.12832 |year=2023 |arxiv=2310.12832 |last2=Kwon |first2=Adrian Wang}}
===Citations===
{{Reflist}}
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