Ordinal collapsing function: Difference between revisions

Revision as of 23:12, 28 June 2023

Revision as of 23:54, 28 June 2023

Line 50:

==== Values of ''ψ'' up to the Feferman–Schütte ordinal ====

The fact that <math>\psi(\Omega+\alpha)</math> equals <math>\varepsilon_{\zeta_0+\alpha}</math> remains true for all <math>\alpha \leq \zeta_1 = \varphi_2phi_2(1)</math>. (Note, in particular, that <math>\psi(\Omega+\zeta_0) = \varepsilon_{\zeta_0\cdot2}</math>: but since now the ordinal <math>\zeta_0</math> has been constructed there is nothing to prevent from going beyond this). However, at <math>\zeta_1 = \phi_2(1)</math> (the first fixed point of <math>\alpha\mapsto \varepsilon_\alpha</math> beyond <math>\zeta_0</math>), the construction stops again, because <math>\zeta_1</math> cannot be constructed from smaller ordinals and <math>\zeta_0</math> by finitely applying the <math>\varepsilon</math> function. So we have <math>\psi(\Omega^2) = \zeta_1</math>.

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>.