Veblen function: Difference between revisions

Revision as of 18:16, 22 August 2016

Revision as of 21:21, 29 May 2017

Line 42:

===Finitely many variables===

For the building of Veblen function with arbitrary finitelу amount of arguments (finitary Veblen function), let's consider <math>\varphi_\alpha(\gamma)</math> as binary function <math>\varphi(\alpha, \gamma)</math>.

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, namely:

Let ''z'' is empty string or a string with one or more zeros <math>0,0,...,0</math> and ''s'' is empty string or an arbitrary string of ordinal variables <math>\alpha_1, \alpha_2,...,\alpha_n</math> with <math>\alpha_1>0</math>. Binary function <math>\varphi(\alpha, \gamma)</math> can be written as <math>\varphi(s,\alpha, z,\gamma)</math> where both ''s'' and ''z'' are empty strings.

* φ(α)=ω^α for a single variable,

* φ(0,α_''n''−1,…,α₀)=φ(α_''n''−1,…,α₀), and

The extended Veblen functions are defined as follows:

* for α>0, γ↦φ(α_''n'',…,α_''i''+1,α,0,…,0,γ) is the function enumerating the common fixed points of the functions ξ↦φ(α_''n'',…,α_''i''+1,β,ξ,0,…,0) for all β&lt;α.

*<math>\varphi(\gamma)=\omega^\gamma</math>,

*<math>\varphi(z,s,\gamma)=\varphi(s,\gamma)</math>,

* if <math>\alpha_{n+1}>0</math>, where <math>n\in \mathbb N_0</math>, then <math>\varphi(s,\alpha_{n+1}, z, \gamma)</math> denotes the <math>\gamma</math>th common fixed point of the functions <math>\xi \mapsto \varphi(s, \beta, \xi,z)</math> for each <math>\beta<\alpha_{n+1}</math>.

For example, φ(1,0,γ) is the γ-th fixed point of the functions ξ↦φ(ξ,0), namely Γ_γ; then φ(1,1,γ) enumerates the fixed points of that function, i.e., of the ξ↦Γ_ξ function; and φ(2,0,γ) enumerates the fixed points of all the ξ↦φ(1,ξ,0). 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 ordinal φ(1,0,0,0) is sometimes known as the [[Ackermann ordinal]]. The limit of the φ(1,0,…,0) where the number of zeroes ranges over ω, is sometimes known as the [[Small Veblen ordinal|“small” Veblen ordinal]].

Every non-zero ordinal <math>\alpha</math> less than the [[small Veblen ordinal]] (SVO) can be uniquely written in normal form for the finitary Veblen function:

<math> \alpha=\varphi(s_1)+\varphi(s_2)+\cdots+\varphi(s_k)</math>

where

*<math>\varphi(s_1)\geq\varphi(s_2)\geq\cdots\geq\varphi(s_k)</math>,

*<math>s_m</math> is an arbitrary string of ordinal variables <math>\alpha_{m,1}, \alpha_{m,2},...,\alpha_{m,n_m}</math> for <math>m \in \{1,...,k\}</math>

* <math>\alpha_{m,1}>0</math> and <math>\alpha_{m,i} <\varphi(s_m)</math> for <math>m \in \{1,...,k\}</math> and <math>i \in \{1,..,n_m\}</math>,

* <math>k, n_1,...,n_k</math> are positive integers.

=== Fundamental sequences for limit ordinals of finitary Veblen function ===

For limit ordinals <math>\alpha<SVO</math>, written in normal form for the finitary Veblen function

2.1) <math>(\varphi(s_1)+\varphi(s_2)+\cdots+\varphi(s_k))[n]=\varphi(s_1)+\varphi(s_2)+\cdots+\varphi(s_k)[n]</math>,

2.2) <math>\varphi(\gamma)[n]=\left\{\begin{array}{lcr}

n \quad \text{if} \quad \gamma=1\\

\varphi(\gamma-1)\cdot n \quad \text{if} \quad \gamma \quad \text{is a successor ordinal}\\

\varphi(\gamma[n]) \quad \text{if} \quad \gamma \quad \text{is a limit ordinal}\\

\end{array}\right.

</math>,

2.3) <math>\varphi(s,\beta,z,\gamma)[0]=0</math> and

<math>\varphi(s,\beta,z,\gamma)[n+1]=\varphi(s,\beta-1,\varphi(s,\beta,z,\gamma)[n],z)</math> if <math>\gamma=0</math> and <math>\beta</math> is a successor ordinal,

2.4) <math>\varphi(s,\beta,z,\gamma)[0]=\varphi(s,\beta,z,\gamma-1)+1</math> and <math>\varphi(s,\beta,z,\gamma)[n+1]=\varphi(s,\beta-1,\varphi(s,\beta,z,\gamma)[n],z)</math> if <math>\gamma</math> and <math>\beta</math> are successor ordinals,

2.5) <math>\varphi(s,\beta,z,\gamma)[n]=\varphi(s,\beta,z,\gamma[n])</math> if <math>\gamma</math> is a limit ordinal,

2.6) <math>\varphi(s,\beta,z,\gamma)[n]=\varphi(s,\beta[n],z,\gamma)</math> if <math>\gamma=0</math> and <math>\beta</math> is a limit ordinal,

2.7) <math>\varphi(s,\beta,z,\gamma)[n]=\varphi(s,\beta[n],\varphi(s,\beta,z,\gamma-1)+1,z)</math> if <math>\gamma</math> is a successor ordinal and <math>\beta</math> is a limit ordinal.

===Transfinitely many variables===