Ordinal collapsing function: Difference between revisions

Revision as of 20:50, 25 September 2022

Revision as of 20:51, 25 September 2022

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Here, <math>\Omega_2</math> is a new ordinal guaranteed to be greater than all the ordinals which will be constructed using <math>\psi_1</math>: again, letting <math>\Omega = \omega_1</math> and <math>\Omega_2 = \omega_2</math> works.

For example, <math>\psi_1(0) = \Omega</math>, and more generally <math>\psi_1(\alpha) = \varepsilon_{\Omega+\alpha}</math> for all countable ordinals and even beyond (<math>\psi_1(\Omega) = \psi_1(\psi_1(0)) = \varepsilon_{\Omega 2}</math> and <math>\psi_1(\psi_1(1)) = \varepsilon_{\varepsilon_{\Omega+1}}</math>): this holds up to the first fixed point <math>\zeta_{\Omega+1}</math> of the function $<math>\xi\mapsto\varepsilon_\xi</math> beyond <math>\Omega</math>, which is the limit of <math>\psi_1(0)</math>, <math>\psi_1(\psi_1(0))</math> and so forth. Beyond this, we have <math>\psi_1(\alpha) = \zeta_{\Omega+1}</math> and this remains true until <math>\Omega_2</math>: exactly as was the case for <math>\psi(\Omega)</math>, we have <math>\psi_1(\Omega_2) = \zeta_{\Omega+1}</math> and <math>\psi_1(\Omega_2+1) = \varepsilon_{\zeta_{\Omega+1}+1}</math>.

The <math>\psi_1</math> function gives us a system of notations (''assuming'' we can somehow write down all countable ordinals!) for the uncountable ordinals below <math>\psi_1(\varepsilon_{\Omega_2+1})</math>, which is the limit of <math>\psi_1(\Omega_2)</math>, <math>\psi_1({\Omega_2}^{\Omega_2})</math> and so forth.