In mathematics, the Veblen functions are a hierarchy of normal function

Normal function

In axiomatic set theory, a function f : Ord → Ord is called normal iff it is continuous and strictly monotonically increasing. This is equivalent to the following two conditions:...

s (continuous

Continuous function (set theory)

In mathematics, specifically set theory, a continuous function is a sequence of ordinals such that the values assumed at limit stages are the limits of all values at previous stages...

 strictly increasing function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

s from ordinal

Ordinal number

In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...

s to ordinals), introduced by Oswald Veblen

Oswald Veblen

Oswald Veblen was an American mathematician, geometer and topologist, whose work found application in atomic physics and the theory of relativity. He proved the Jordan curve theorem in 1905.-Life:...

 in . If φ₀ is any normal function, then for any non-zero ordinal α, φ_α is the function enumerating the common fixed point

Fixed point (mathematics)

In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...

s of φ_β for β<α. These functions are all normal.

The Veblen hierarchy

In the special case when φ₀(α)=ω^α
this family of functions is known as the Veblen hierarchy.
The function φ₁ is the same as the ε function: φ₁(α)= ε_α. If then 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 ).

Fundamental sequences for the Veblen hierarchy

The fundamental sequence for an ordinal with cofinality

Cofinality

In mathematics, especially in order theory, the cofinality cf of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A....

 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.

The Γ function

The function Γ enumerates the ordinals α such that φ_α(0) = α.
Γ₀ is the Feferman–Schütte ordinal

Feferman–Schütte ordinal

In mathematics, the Feferman–Schütte ordinal Γ0 is a large countable ordinal.It is the proof theoretic ordinal of several mathematical theories, such as arithmetical transfinite recursion.It is named after Solomon Feferman and Kurt Schütte....

, i.e. it is the smallest α such that φ_α(0) = α.

For Γ₀, a fundamental sequence could be chosen to be and

For Γ_β+1, let and

For Γ_β where is a limit, let

Generalizations

In this section it is more convenient to think of φ_α(β) as a function φ(α,β) of two variables. Veblen showed how to generalize the definition to produce a function φ(α_n,α_n-1, ...,α₀) of several variables. More generally he 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. So one is defining a function φ from κ^κ into κ.