Veblen function: Difference between revisions

Revision as of 22:23, 29 May 2017

Revision as of 03:48, 30 May 2017

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===Finitely many variables===

ForTo build the building of Veblen function withof arbitrarya finite amountnumber of arguments (finitary Veblen functions), let's considerthe binary function <math>\varphi_varphi(\alpha(, \gamma)</math> as binary functionbe <math>\varphi(varphi_\alpha, (\gamma)</math>. as defined above.

Let ''z'' be an empty string or a string with one or more zeros <math>0,0,...,0</math> and ''s'' be an empty string or an arbitrary string of ordinal variablesordinals <math>\alpha_1, \alpha_2,...,\alpha_n</math> with <math>\alpha_1>0</math>. The 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.

The finitary Veblen functions are defined as follows:

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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 variablesordinals <math>\alpha_{m,1}, \alpha_{m,2},...,\alpha_{m,n_m}</math> smaller than the SVO 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.