Veblen function

In mathematics, the Veblen functions are a hierarchy of functions from ordinals to ordinals, introduced by harvtxt|Veblen|1908. If φ₀ 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 φ₀(α)=ω^αthis family of functions is known as the Veblen hierarchy. The function φ₁ is the same as the ε function: φ₁(α)= ε_α.Ordering: $varphi_alpha(eta) < varphi_gamma(delta)$ if and only if either ( $alpha = gamma$ and $eta < delta$ ) or ( $alpha < gamma$ and $eta < varphi_gamma(delta)$ ) or ( $alpha > gamma$ and $varphi_alpha(eta) < delta$ ).

Fundamental sequences for the Veblen hierarchy

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 $varphi_{eta_1}(gamma_1) + varphi_{eta_2}(gamma_2) + cdots + varphi_{eta_k}(gamma_k)$ , 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 $varphi_{eta_j}$ . 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 $varphi_0(0)$ = ω⁰ = 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 $varphi_{eta+1}(0)$ , we use 0, $varphi_{eta}(0)$ , $varphi_{eta}(varphi_{eta}(0))$ , $varphi_{eta}(varphi_{eta}(varphi_{eta}(0)))$ , etc..

For $varphi_{eta+1}(gamma+1)$ , we use $varphi_{eta+1}(gamma)+1$ , $varphi_{eta}(varphi_{eta+1}(gamma)+1)$ , $varphi_{eta}(varphi_{eta}(varphi_{eta+1}(gamma)+1))$ , etc..

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

Now suppose that β is a limit:If $eta < varphi_{eta}(0)$ , then for $varphi_{eta}(0)$ , we replace β by its fundamental sequence.

For $varphi_{eta}(gamma+1)$ , use $varphi_{eta_m}(varphi_{eta}(gamma)+1)$ where $eta_m$ is the fundamental sequence for β.

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

Otherwise, the ordinal cannot be described in terms of smaller ordinals using $varphi$ 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-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", ...,α₀) of several variables. More generally he showed thatφ can be defined even for a transfinite sequence of ordinals α_β, provided that all but a finite number of them are zero.

References

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