Ordinal collapsing function: Difference between revisions

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Revision as of 16:40, 15 February 2013

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==== Beyond the Feferman-Schütte ordinal ====

We have <math>\psi(\Omega^{\Omega+\Omega^\alpha}) = \phi_{\Gamma_0+\alpha}(0)</math> for all <math>\alpha\leq\Gamma_1</math> where <math>\Gamma_1</math> is the next fixed point of <math>\alpha \mapsto \phi_\alpha(0)</math>. So, if <math>\alpha\mapsto\Gamma_\alpha</math> enumerates the fixed points in question. (which can also be noted <math>\phi(\alpha1,0,0\alpha)</math> using the many-valued Veblen functions) we have <math>\psi(\Omega^{\Omega(1+\alpha)}) = \Gamma_\alpha</math>, until the first fixed point <math>\phi(1,1,0)</math> of the <math>\alpha\mapsto\Gamma_\alpha</math> itself, which will be <math>\psi(\Omega^{\Omega^+1})</math> (and the first fixed point <math>\phi(2,0,0)</math> of the <math>\alpha \mapsto \phi(1,\alpha,0)</math> functions will be <math>\psi(\Omega^{\Omega2})</math>). In this manner:

* <math>\psi(\Omega^{\Omega^2})</math> is the [[Ackermann ordinal]] (the range of the notation <math>\phi(\alpha,\beta,\gamma)</math> defined predicatively),

* <math>\psi(\Omega^{\Omega^\omega})</math> is the [[small Veblen ordinal|&ldquo;small&rdquo; Veblen ordinal]] (the range of the notations <math>\phi(\ldots)</math> predicatively using finitely many variables),