Ordinal collapsing function: Difference between revisions

Revision as of 04:41, 5 August 2022

Revision as of 04:41, 5 August 2022

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You can also have <math>\Omega</math> or more cardinals, in fact as many as nesting of the &psi; function allows:

:<math>\psi_\nu(\alpha)</math> is the smallest ordinal which cannot be expressed from <math>0</math>, <math>1</math>, <math>\omega</math>, and all ordinals smaller than <math>\Omega_\nu</math> using sums, products, exponentials, and the <math>\psi_\kappa</math> functions (for <math>\kappa</math> being a previously constructed ordinal) to previously constructed ordinals less than <math>\alpha</math>.

<math>\Omega_0</math> being 0 here, abbreviate <math>\psi_0</math> as <math>\psi</math>. The limit of this system is <math>\psi(\text{The first ordinal }\alpha\text{ such that }\Omega_\alpha = \alpha})</math>

=== A "normal" variant ===