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 φ0 is any normal function, then for any non-zero ordinal α, φα is the function enumerating the common fixed points of φβ for β<α.
In the special case when φ0(α)=ωα this family of functions is known as the Veblen hierarchy.
[2] From this and the fact that φβ is strictly increasing we get the ordering:
[2] The fundamental sequence for an ordinal with cofinality ω is a distinguished strictly increasing ω-sequence that 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.
A variation of Cantor normal form used in connection with the Veblen hierarchy is: every nonzero ordinal number α can be uniquely written as
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
The function Γ enumerates the ordinals α such that φα(0) = α. Γ0 is the Feferman–Schütte ordinal, i.e. it is the smallest α such that φα(0) = α.
For Γ0, a fundamental sequence could be chosen to be
φ ( α , γ )
φ ( β , γ )
φ ( s , β , z , γ )
The finitary Veblen functions are defined as follows: For example,
enumerates the fixed points of that function, i.e., of 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).
where the number of zeroes ranges over ω, is sometimes known as the "small" Veblen ordinal.
less than the small Veblen ordinal (SVO) can be uniquely written in normal form for the finitary Veblen function:
, written in normal form for the finitary Veblen function: More generally, Veblen showed that φ can be defined even for a transfinite sequence of ordinals αβ, 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 κκ (ordinal exponentiation).
So one is defining a function φ from κκ 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) that ends in zero (i.e., such that α0=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 that 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@ι0,ξ@ι] meaning that for the smallest index ι0 such that αι0 is nonzero the latter has been replaced by some value ζ<αι0 and that for some smaller index ι<ι0, the value αι=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., that 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.
The function takes on several prominent values: