User:Gro-Tsen/An ordinal collapsing function

This is an attempt to define and explicit a not-too-complicated ordinal collapsing function which should be useful for pedagogical purposes (to construct large countable ordinals). It presumably defines an ordinal notation up to the Bachmann-Howard ordinal, but I need to check this.

Definition

Let $\Omega$ stand for the first uncountable ordinal, or, in fact, any ordinal guaranteed to be greater than all the countable ordinals which will be constructed (for example, the Church-Kleene ordinal is adequate for our purposes).

We define a function $\psi$, taking an arbitrary ordinal $\alpha$ to a countable ordinal $\psi (\alpha )$, recursively on $\alpha$, as follows:

Assume $\psi (\beta )$ has been defined for all $\beta <\alpha$, and we wish to define $\psi (\alpha )$.

Let $C(\alpha )$ be the set of ordinals generated starting from $0$, $1$, $\omega$ and $\Omega$ by recursively applying the following functions: ordinal addition, multiplication and exponentiation and the function $\psi \upharpoonright _{\alpha }$, i.e., the restriction of $\psi$ to ordinals $\beta <\alpha$. (Formally, we define $C(\alpha )_{0}=\{0,1,\omega ,\Omega \}$ and inductively $C(\alpha )_{n+1}=C(\alpha )_{n}\cup \{\beta _{1}+\beta _{2},\beta _{1}\beta _{2},{\beta _{1}}^{\beta _{2}}:\beta _{1},\beta _{2}\in C(\alpha )_{n}\}\cup \{\psi (\beta ):\beta \in C(\alpha )_{n}\land \beta <\alpha \}$ for all integers $n$ and we let $C(\alpha )$ be the union of the $C(\alpha )_{n}$ for all $n$.)

Then $\psi (\alpha )$ is defined as the smallest ordinal not belonging to $C(\alpha )$.

Examples

First consider $C(0)$. It contains ordinals $0$, $1$, $2$, $3$, $\omega$, $\omega +1$, $\omega +2$, $\omega 2$, $\omega 3$, $\omega ^{2}$, $\omega ^{3}$, $\omega ^{\omega }$, $\omega ^{\omega ^{\omega }}$ and so on. It also contains such ordinals as $\Omega$, $\Omega +1$, $\Omega \omega$, $\Omega ^{\Omega }$. The first ordinal which it does not contain is $\varepsilon _{0}$ (which is the limit of $\omega$, $\omega ^{\omega }$, $\omega ^{\omega ^{\omega }}$ and so on — less than $\Omega$ by assumption). The upper bound of the ordinals it contains is $\varepsilon _{\Omega +1}$ (the limit of $\Omega$, $\Omega ^{\Omega }$, $\Omega ^{\Omega ^{\Omega }}$ and so on), but that is not so important. This shows that $\psi (0)=\varepsilon _{0}$.