Ordinal collapsing function: Difference between revisions

Revision as of 20:28, 18 September 2024

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==== Beyond the Feferman–Schütte ordinal ====

We have <math>\psi(\Omega^\Omega+\Omega^\alpha) = \phi_varphi_{\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_varphi_\alpha(0)</math>. So, if <math>\alpha\mapsto\Gamma_\alpha</math> enumerates the fixed points in question (which can also be noted <math>\phivarphi(1,0,\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>\phivarphi(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>\phivarphi(2,0,0)</math> of the <math>\alpha \mapsto \phivarphi(1,\alpha,0)</math> functions will be <math>\psi(\Omega^{\Omega\cdot 2})</math>). In this manner:

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

* <math>\psi(\Omega^{\Omega^\omega})</math> is the [[small Veblen ordinal|"small" Veblen ordinal]] (the range of the notations <math>\phivarphi(\cdot)</math> predicatively using finitely many variables),

* <math>\psi(\Omega^{\Omega^\Omega})</math> is the [[large Veblen ordinal|"large" Veblen ordinal]] (the range of the notations <math>\phivarphi(\cdot)</math> predicatively using transfinitely-but-predicatively-many variables),

* the limit <math>\psi(\varepsilon_{\Omega+1})</math> of <math>\psi(\Omega)</math>, <math>\psi(\Omega^\Omega)</math>, <math>\psi(\Omega^{\Omega^\Omega})</math>, etc., is the [[Bachmann–Howard ordinal]]: after this our function <math>\psi</math> is constant, and we can go no further with the definition we have given.