Ordinal collapsing function: Difference between revisions

Revision as of 23:00, 28 June 2023

Revision as of 23:01, 28 June 2023

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The same reasoning shows that <math>\psi(\Omega(1+\alpha)) = \phi_2(\alpha)</math> for all <math>\alpha\leq\phi_3(0)</math>, where <math>\phi_2</math> enumerates the fixed points of <math>\phi_1\colon\alpha\mapsto\varepsilon_\alpha</math> and <math>\phi_3(0)</math> is the first fixed point of <math>\phi_2</math>. We then have <math>\psi(\Omega^2) = \phi_3(0)</math>.

Again, we can see that <math>\psi(\Omega^\alpha) = \varphi_phi_{1+\alpha}(0)</math> for some time: this remains true until the first fixed point <math>\Gamma_0</math> of <math>\alpha \mapsto \phi_\alpha(0)</math>, which is the [[Feferman–Schütte ordinal]]. Thus, <math>\psi(\Omega^\Omega) = \Gamma_0</math> is the Feferman–Schütte ordinal.

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