In the special case when φ0(α)=ωα
this family of functions is known as the Veblen hierarchy.
The function φ1 is the same as the ε function: φ1(α)= εα. If then From this and the fact that φβ is strictly increasing we get the ordering: if and only if either ( and ) or ( and ) or ( and ).
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, (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 = 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.
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).
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 (i.e., an ordinal function with finite support) which ends in zero (i.e., such that α₀=0), and let α[0↦γ] denote the same function where the final 0 has been replaced by γ. Then γ↦φ(α[0↦γ]) 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[ι₀↦ζ,ι↦ξ] 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=(ω↦1) denotes the transfinite sequence with value 1 at ω and 0 everywhere else, then φ(ω↦1) 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.