Veblen function: Difference between revisions

Revision as of 04:33, 6 January 2018

Revision as of 21:26, 19 February 2018

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

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

* if <math>\beta >0</math>, then <math>\varphi (s,\beta ,z,\gamma )</math> denotes the <math>(1+\gamma [I think this is wrong because 1+\gamma=\gamma] )</math>-th common fixed point of the functions <math>\xi \mapsto \varphi (s,\delta ,\xi ,z)</math> for each <math>\delta <\beta</math>

For example, <math>\varphi(1,0,\gamma)</math> is the <math>(1+\gamma)</math>-th fixed point of the functions <math>\xi\mapsto\varphi(\xi,0)</math>, namely <math>\Gamma_\gamma</math>; then <math>\varphi(1,1,\gamma)</math> enumerates the fixed points of that function, i.e., of the <math>\xi\mapsto\Gamma_\xi</math> function; and <math>\varphi(2,0,\gamma)</math> enumerates the fixed points of all the <math>\xi\mapsto\varphi(1,\xi,0)</math>. 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).