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In mathematics, the **Veblen functions** are a hierarchy of normal functions (continuous strictly increasing functions from ordinals to ordinals), introduced by Oswald Veblen in Veblen (1908). If φ_{0} is any normal function, then for any non-zero ordinal α, φ_{α} is the function enumerating the common fixed points of φ_{β} for β<α. These functions are all normal.

In the special case when φ_{0}(α)=ω^{α}
this family of functions is known as the **Veblen hierarchy**.
The function φ_{1} is the same as the ε function: φ_{1}(α)= ε_{α}.^{[1]} If then .^{[2]} From this and the fact that φ_{β} is strictly increasing we get the ordering: if and only if either ( and ) or ( and ) or ( and ).^{[2]}

The fundamental sequence for an ordinal with cofinality ω is a distinguished strictly increasing ω-sequence which has the ordinal as its limit. If one has fundamental sequences for α and all smaller limit ordinals, then one can create an explicit constructive bijection between ω and α, (i.e. one not using the axiom of choice). Here we will describe fundamental sequences for the Veblen hierarchy of ordinals. The image of *n* under the fundamental sequence for α will be indicated by α[*n*].

A variation of Cantor normal form used in connection with the Veblen hierarchy is — every nonzero ordinal number α can be uniquely written as , where *k*>0 is a natural number and each term after the first is less than or equal to the previous term, and each If a fundamental sequence can be provided for the last term, then that term can be replaced by such a sequence to get

For any β, if γ is a limit with then let

No such sequence can be provided for = ω^{0} = 1 because it does not have cofinality ω.

For we choose

For we use and i.e. 0, , , etc..

For , we use and

Now suppose that β is a limit:

If , then let

For , use

Otherwise, the ordinal cannot be described in terms of smaller ordinals using and this scheme does not apply to it.

The function Γ enumerates the ordinals α such that φ_{α}(0) = α.
Γ_{0} is the Feferman–Schütte ordinal, i.e. it is the smallest α such that φ_{α}(0) = α.

For Γ_{0}, a fundamental sequence could be chosen to be and

For Γ_{β+1}, let and

For Γ_{β} where is a limit, let

To build the Veblen function of a finite number of arguments (finitary Veblen function), let the binary function be as defined above.

Let be an empty string or a string consisting of one or more comma-separated zeros and be an empty string or a string consisting of one or more comma-separated ordinals with . The binary function can be written as where both and are empty strings. The finitary Veblen functions are defined as follows:

- if , then denotes the -th common fixed point of the functions for each

For example, is the -th fixed point of the functions , namely ; then enumerates the fixed points of that function, i.e., of the function; and enumerates the fixed points of all the . Each instance of the generalized Veblen functions is continuous in the *last nonzero* variable (i.e., if one variable is made to vary and all later variables are kept constantly equal to zero).

The ordinal is sometimes known as the Ackermann ordinal. The limit of the where the number of zeroes ranges over ω, is sometimes known as the "small" Veblen ordinal.

Every non-zero ordinal less than the small Veblen ordinal (SVO) can be uniquely written in normal form for the finitary Veblen function:

where

- is a positive integer
- is a string consisting of one or more comma-separated ordinals where and each

For limit ordinals , written in normal form for the finitary Veblen function:

- ,
- ,
- and if and is a successor ordinal,
- and if and are successor ordinals,
- if is a limit ordinal,
- if and is a limit ordinal,
- if is a successor ordinal and is a limit ordinal.

More generally, Veblen showed that φ can be defined even for a transfinite sequence of ordinals α_{β}, provided that all but a finite number of them are zero. Notice that if such a sequence of ordinals is chosen from those less than an uncountable regular cardinal κ, then the sequence may be encoded as a single ordinal less than κ^{κ} (ordinal exponentiation). So one is defining a function φ from κ^{κ} into κ.

The definition can be given as follows: let __α__ be a transfinite sequence of ordinals (i.e., an ordinal function with finite support) *which ends in zero* (i.e., such that α_{0}=0), and let __α__[γ@0] denote the same function where the final 0 has been replaced by γ. Then γ↦φ(__α__[γ@0]) is defined as the function enumerating the common fixed points of all functions ξ↦φ(__β__) where __β__ ranges over all sequences which are obtained by decreasing the smallest-indexed nonzero value of __α__ and replacing some smaller-indexed value with the indeterminate ξ (i.e., __β__=__α__[ζ@ι_{0},ξ@ι] meaning that for the smallest index ι_{0} such that α_{ι0} is nonzero the latter has been replaced by some value ζ<α_{ι0} and that for some smaller index ι<ι_{0}, the value α_{ι}=0 has been replaced with ξ).

For example, if __α__=(1@ω) 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, or "great" Veblen number.^{[3]}

In 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.

The function takes on several prominent values:

- is the proof-theoretic ordinal of Peano arithmetic and the limit of what ordinals can be represented in terms of Cantor normal form and smaller ordinals.
- , a bound on the order types of the recursive path orderings with finitely many function symbols, and the smallest ordinal closed under primitive recursive ordinal functions.
^{[4]}^{[5]} - The Feferman-Schutte ordinal is equal to .
^{[6]} - The small Veblen ordinal is equal to .
^{[7]}

- Hilbert Levitz,
*Transfinite Ordinals and Their Notations: For The Uninitiated*, expository article (8 pages, in PostScript) - Pohlers, Wolfram (1989),
*Proof theory*, Lecture Notes in Mathematics, vol. 1407, Berlin: Springer-Verlag, doi:10.1007/978-3-540-46825-7, ISBN 978-3-540-51842-6, MR 1026933 - Schütte, Kurt (1977),
*Proof theory*, Grundlehren der Mathematischen Wissenschaften, vol. 225, Berlin-New York: Springer-Verlag, pp. xii+299, ISBN 978-3-540-07911-8, MR 0505313 - Takeuti, Gaisi (1987),
*Proof theory*, Studies in Logic and the Foundations of Mathematics, vol. 81 (Second ed.), Amsterdam: North-Holland Publishing Co., ISBN 978-0-444-87943-1, MR 0882549 - Smorynski, C. (1982), "The varieties of arboreal experience",
*Math. Intelligencer*,**4**(4): 182–189, doi:10.1007/BF03023553 contains an informal description of the Veblen hierarchy. - Veblen, Oswald (1908), "Continuous Increasing Functions of Finite and Transfinite Ordinals",
*Transactions of the American Mathematical Society*,**9**(3): 280–292, doi:10.2307/1988605, JSTOR 1988605 - Miller, Larry W. (1976), "Normal Functions and Constructive Ordinal Notations",
*The Journal of Symbolic Logic*,**41**(2): 439–459, doi:10.2307/2272243, JSTOR 2272243 - Massmann, Jayde Sylvie; Kwon, Adrian Wang (October 20, 2023),
*Extending the Veblen Function*, arXiv:2310.12832`{{citation}}`

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**^**Stephen G. Simpson,*Subsystems of Second-order Arithmetic*(2009, p.387)- ^
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