In mathematics, the Veblen functions are a hierarchy of normal functions, introduced by Oswald Veblen in. 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. These functions are all normal.
The Veblen hierarchy
In the special case when φ0=ωα this family of functions is known as the Veblen hierarchy. The function φ1 is the same as the ε function: φ1= εα. If then From this and the fact that φβ is strictly increasing we get the ordering: if and only if either or or.
Fundamental sequences for the Veblen hierarchy
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,. Here we will describe fundamental sequences for the Veblen hierarchy of ordinals. The image ofn under the fundamental sequence for α will be indicated by α. 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 = ω0 = 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 Γ function
The function Γ enumerates the ordinals α such that φα = α. Γ0 is the Feferman–Schütte ordinal, i.e. it is the smallest α such that φα = α. For Γ0, a fundamental sequence could be chosen to be and For Γβ+1, let and For Γβ where is a limit, let
To build the Veblen function of a finite number of arguments, 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. 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 can be uniquely written in normal form for the finitary Veblen function: where
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. 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 which ends in zero, and let α denote the same function where the final 0 has been replaced by γ. Then γ↦φ is defined as the function enumerating the common fixed points of all functions ξ↦φ 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. For example, if α= denotes the transfinite sequence with value 1 at ω and 0 everywhere else, then φ is the smallest fixed point of all the functions ξ↦φ with finitely many final zeroes. The smallest ordinal α such that α is greater than φ applied to any function with support in α is sometimes known as the “large” Veblen ordinal.
