In mathematics, the Veblen functions are a hierarchy of normal functions (continuous strictly increasing functions from ordinals to ordinals), introduced by Oswald Veblen in . If ÃÂ<sub>0</sub> is any normal function, then for any non-zero ordinal ñ, ÃÂ<sub>ñ</sub> is the function enumerating the common fixed points of ÃÂ<sub>ò</sub> for ò<ñ. These functions are all normal.
In the special case when ÃÂ<sub>0</sub>(ñ)=ÃÂ<sup>ñ</sup> this family of functions is known as the Veblen hierarchy. The function ÃÂ<sub>1</sub> is the same as the õ function: ÃÂ<sub>1</sub>(ñ)= õ<sub>ñ</sub>. If then . From this and the fact that ÃÂ<sub>ò</sub> is strictly increasing we get the ordering: if and only if either ( and ) or ( and ) or ( and ).
The fundamental sequence for an ordinal with cofinality àis a distinguished strictly increasing ÃÂ-sequence that has the ordinal as its limit. If one has fundamental sequences for ñ and all smaller limit ordinals, then one can create an explicit constructive bijection between àand ñ, (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 ñ will be indicated by ñ[n].
A variation of Cantor normal form used in connection with the Veblen hierarchy is: every nonzero ordinal number ñ 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 ò, if ó is a limit with then let
No such sequence can be provided for = ÃÂ<sup>0</sup> = 1 because it does not have cofinality ÃÂ.
For we choose
For we use and i.e. 0, , , etc..
For , we use and
Now suppose that ò 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 àenumerates the ordinals ñ such that ÃÂ<sub>ñ</sub>(0) = ñ. ÃÂ<sub>0</sub> is the FefermanâÂÂSchütte ordinal, i.e. it is the smallest ñ such that ÃÂ<sub>ñ</sub>(0) = ñ.
For ÃÂ<sub>0</sub>, a fundamental sequence could be chosen to be and
For ÃÂ<sub>ò+1</sub>, let and
For ÃÂ<sub>ò</sub> 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:
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 limit of the where the number of zeroes ranges over ÃÂ, 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
For limit ordinals , written in normal form for the finitary Veblen function:
More generally, Veblen showed that àcan be defined even for a transfinite sequence of ordinals ñ<sub>ò</sub>, 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 ú, then the sequence may be encoded as a single ordinal less than ú<sup>ú</sup> (ordinal exponentiation). So one is defining a function àfrom ú<sup>ú</sup> into ú.
The definition can be given as follows: let <u>ñ</u> be a transfinite sequence of ordinals (i.e., an ordinal function with finite support) that ends in zero (i.e., such that ñ<sub>0</sub>=0), and let <u>ñ</u>[ó@0] denote the same function where the final 0 has been replaced by ó. Then óâ¦ÃÂ(<u>ñ</u>[ó@0]) is defined as the function enumerating the common fixed points of all functions þâ¦ÃÂ(<u>ò</u>) where <u>ò</u> ranges over all sequences that are obtained by decreasing the smallest-indexed nonzero value of <u>ñ</u> and replacing some smaller-indexed value with the indeterminate þ (i.e., <u>ò</u>=<u>ñ</u>[ö@ù<sub>0</sub>,þ@ù] meaning that for the smallest index ù<sub>0</sub> such that ñ<sub>ù<sub>0</sub></sub> is nonzero the latter has been replaced by some value ö<ñ<sub>ù<sub>0</sub></sub> and that for some smaller index ù<ù<sub>0</sub>, the value ñ<sub>ù</sub>=0 has been replaced with þ).
For example, if <u>ñ</u>=(1@ω) denotes the transfinite sequence with value 1 at àand 0 everywhere else, then ÃÂ(1@ÃÂ) is the smallest fixed point of all the functions þâ¦ÃÂ(þ,0,...,0) with finitely many final zeroes (it is also the limit of the ÃÂ(1,0,...,0) with finitely many zeroes, the small Veblen ordinal).
The smallest ordinal ñ such that ñ is greater than àapplied to any function with support in ñ (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.
Here is a simplified version of the transfinitary Veblen function:
We modify this to use functions (of finite support) from the class of all ordinals to itself as input. Such functions can be encoded by a set (rather than a proper class) as follows:
Using s and t for such sets and α, β, γ, and δ for ordinals, the definitions are:
Notations in this system begin with the zero-ary function 0, and use the binary function addition + to combine powers of ω which come from applying the transfinite Veblen function φ to these set-coded functions.
et cetera.
How many times can φ take on some value? The values are always powers of ω.
So powers of ω are always values of φ at least once. Epsilon numbers are fixed points of that, so they are always values at least twice. Some ordinals are values of φ infinitely often. For example, Ω is a value an uncountable number of times.
So if D<sub>φ</sub> < φ, then that is the last time that φ can take that value. So D<sub>φ</sub> = φ is the norm for ordinals which are values more than once. D<sub>φ</sub> > φ never occurs. But
and there are many other ordinals for which that is true. They are all very strong limits.
The preferred form of s to produce a value of φ (s) is the one for which D<sub>φ</sub> (s) < φ (s). The ordering theorem is:
If α is an ordinal with no preferred form, then:
The ordering among epsilon numbers which do not have a preferred form depends on how those epsilon numbers are specified. To give us a way to talk about them, let np<sub>α</sub> be an enumeration of the epsilon numbers lacking a preferred form. Let us also define the complexity, X, of ordinals less than np<sub>ω</sub> as follows:
The complexity is a (finite) natural number whenever it is defined. And for any natural number k there are only finitely many ordinals α for which X (α) takes the value k or less.
The fundamental sequence of 0 is the empty sequence. The fundamental sequence of a successor ordinal is (α+1) [0] = α. For ordinals with cofinality âÂÂ¥ ω written in normal form with length k > 1 and k < ω, the fundamental sequence is:
If φ (s) has a preferred form and is less than np<sub>ω</sub>, that is it must be strictly greater than D<sub>φ</sub> (s) :
then we can define the fundamental sequence of φ (s) by (φ (s)) [n] = ρ<sub>n</sub> where:
We will show that φ (s) = ρ<sub>ω</sub>. First notice that this sequence is strictly increasing because
Thus
On the other hand,
but φ (s) â 0, 1, or 2, since it has cofinality âÂÂ¥ ω and φ (s) â D<sub>φ</sub> (s) since it has a preferred form. Thus
Now, let us use mathematical induction with the inductive hypothesis ρ<sub>n</sub> < φ (s) : suppose
thus
Otherwise, φ (s) lacks a preferred form and we can set:
via transfinite recursion. This sequence may be longer than ω, but that is unavoidable since the Veblen functions are unable to comprehend some of the more complex ways of forming cofinal sequences. That is, such ordinals include (and may be indistinguishable from) uncountable regular ordinals which do not have fundamental sequences of length ω.
In , 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 ÃÂ(1@(1,0)) are valid (representing the large Veblen ordinal), visualised as multi-dimensional arrays. It was proven that all ordinals below the BachmannâÂÂHoward 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: