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In mathematics, the Veblen functions are a hierarchy of functions from ordinals to ordinals, introduced by . If φ_{0} is any continuous strictly increasing function from ordinals to ordinals, then for any non-zero ordinal α, φ_{α} is the function enumerating the common fixed points of φ_{β} for β<α. These functions are all continuous strictly increasing functions (i.e. normal functions) from ordinals to ordinals.
## The Veblen hierarchy

In the special case when φ_{0}(α)=ω^{α}
this family of functions is known as the Veblen hierarchy.
The function φ_{1} is the same as the ε function: φ_{1}(α)= ε_{α}.
Ordering: $varphi\_alpha(beta)\; <\; varphi\_gamma(delta)$ if and only if either ($alpha\; =\; gamma$ and $beta\; <\; delta$) or ($alpha\; <\; gamma$ and $beta\; <\; varphi\_gamma(delta)$) or ($alpha\; >\; gamma$ and $varphi\_alpha(beta)\; <\; delta$).
### Fundamental sequences for the Veblen hierarchy

### The Γ function

The function Γ enumerates the ordinals α such that φ_{α}(0) = α.
Γ_{0} is the Feferman-Schutte ordinal, i.e. it is the smallest α such that φ_{α}(0) = α.
## 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_{0}) 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.
## References

The fundamental sequence of 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.

A variation of Cantor normal form used in connection with the Veblen hierarchy is -- every ordinal number α can be uniquely written as $varphi\_\{beta\_1\}(gamma\_1)\; +\; varphi\_\{beta\_2\}(gamma\_2)\; +\; cdots\; +\; varphi\_\{beta\_k\}(gamma\_k)$, where k is a natural number and each term after the first is less than or equal to the previous term and each $gamma\_j$ is not a fixed point of $varphi\_\{beta\_j\}$. If a fundamental sequence can be provided for the last term, then that term can be replaced by such a sequence to get a fundamental sequence for α.

No such sequence can be provided for $varphi\_0(0)$ = ω^{0} = 1 because it does not have cofinality ω.

For $varphi\_0(gamma+1)\; =\; omega\; ^\{gamma+1\}\; =\; omega^\; gamma\; cdot\; omega$, we choose the function which maps the natural number m to $omega^gamma\; cdot\; m$.

If γ is a limit which is not a fixed point of $varphi\_0$, then for $varphi\_0(gamma)$, we replace γ by its fundamental sequence inside $varphi\_0$.

For $varphi\_\{beta+1\}(0)$, we use 0, $varphi\_\{beta\}(0)$, $varphi\_\{beta\}(varphi\_\{beta\}(0))$, $varphi\_\{beta\}(varphi\_\{beta\}(varphi\_\{beta\}(0)))$, etc..

For $varphi\_\{beta+1\}(gamma+1)$, we use $varphi\_\{beta+1\}(gamma)+1$, $varphi\_\{beta\}(varphi\_\{beta+1\}(gamma)+1)$, $varphi\_\{beta\}(varphi\_\{beta\}(varphi\_\{beta+1\}(gamma)+1))$, etc..

If γ is a limit which is not a fixed point of $varphi\_\{beta+1\}$, then for $varphi\_\{beta+1\}(gamma)$, we replace γ by its fundamental sequence inside $varphi\_\{beta+1\}$.

Now suppose that β is a limit: If $beta\; <\; varphi\_\{beta\}(0)$, then for $varphi\_\{beta\}(0)$, we replace β by its fundamental sequence.

For $varphi\_\{beta\}(gamma+1)$, use $varphi\_\{beta\_m\}(varphi\_\{beta\}(gamma)+1)$ where $beta\_m$ is the fundamental sequence for β.

If γ is a limit which is not a fixed point of $varphi\_\{beta\}$, then for $varphi\_\{beta\}(gamma)$, we replace γ by its fundamental sequence inside $varphi\_\{beta\}$.

Otherwise, the ordinal cannot be described in terms of smaller ordinals using $varphi$ and this scheme does not apply to it.

- Hilbert Levitz, Transfinite Ordinals and Their Notations: For The Uninitiated, expository article (8 pages, in PostScript)

|series= Lecture Notes in Mathematics|volume= 1407|publisher= Springer-Verlag|place= Berlin|year= 1989|ISBN= 3-540-51842-8 }}

- Smorynski, C. The varieties of arboreal experience Math. Intelligencer 4 (1982), no. 4, 182-189; contains an informal description of the Veblen hierarchy.
- Larry W. Miller, Normal Functions and Constructive Ordinal Notations,The Journal of Symbolic Logic,volume 41,number 2,June 1976,pages 439 to 459.

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