Ordinal collapsing function: Difference between revisions

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Similarly, <math>C(1)</math> contains the ordinals which can be formed from <math>0</math>, <math>1</math>, <math>\omega</math>, <math>\Omega</math> and this time also <math>\varepsilon_0</math>, using addition, multiplication and exponentiation. This contains all the ordinals up to <math>\varepsilon_1</math> but not the latter, so <math>\psi(1) = \varepsilon_1</math>. In this manner, we prove that <math>\psi(\alpha) = \varepsilon_\alpha</math> inductively on <math>\alpha</math>: the proof works, however, only as long as <math>\alpha<\varepsilon_\alpha</math>. We therefore have:

:<math>\psi(\alpha) = \varepsilon_\alpha = \phi_1varphi_1(\alpha)</math> for all <math>\alpha\leq\zeta_0</math>, where <math>\zeta_0 = \phi_2varphi_2(0)</math> is the smallest fixed point of <math>\alpha \mapsto \varepsilon_\alpha</math>.

(Here, the <math>\phivarphi</math> functions are the [[Veblen function]]s defined starting with <math>\phi_1varphi_1(\alpha) = \varepsilon_\alpha</math>.)

Now <math>\psi(\zeta_0) = \zeta_0</math> but <math>\psi(\zeta_0+1)</math> is no larger, since <math>\zeta_0</math> cannot be constructed using finite applications of <math>\phi_1varphi_1\colon \alpha\mapsto\varepsilon_\alpha</math> and thus never belongs to a <math>C(\alpha)</math> set for <math>\alpha\leq\Omega</math>, and the function <math>\psi</math> remains "stuck" at <math>\zeta_0</math> for some time:

:<math>\psi(\alpha) = \zeta_0</math> for all <math>\zeta_0 \leq \alpha \leq \Omega</math>.