User blog:P進大好きbot/New Notation up to Church-Kleene Ordinal

I introduced a generic system analogous to Kleene's \(\mathcal{O}\) in my last blog post. This time, I introduce a simpler system which is not analogous to Kleene's \(\mathcal{O}\). I fix an enumeration of Turing machines. I denote by \begin{eqnarray*} f \colon \omega_1^{\textrm{CK}} \to \mathbb{N} \end{eqnarray*} the function which assigns to each \(\alpha \in \omega_1^{\textrm{CK}}\) the minimum of an \(a \in \mathbb{N}\) such that \(a\) is a code of a Turing machine which computes a recursive well-ordering on \(\mathbb{N)\) whose ordinal type is \(\alpha\).

I denote by \(OT \subset \mathbb{N}\) the image of \(f\), and by \begin{eqnarray*} \textrm{ot} \colon OT \to \omega_1^{\textrm{CK}} \end{eqnarray*} the map which assigns to each \(a \in OT\) the ordinal type of the recursive well-ordering on \(\mathbb{N}\) computed by the Turing machine of which \(a\) is a code.

I have obtained a notation \(OT\) admitting a bijective map \(\textrm{ot}\) to \(\omega_1^{\textrm{CK}}\). It is quite simple. If you need a well-ordering, then you can simply use the binary relation \(a <_{OT} b\) on \(OT\) defined as \(\textrm{ot}(a) \in \textrm{ot}(b)\).

In order to create a large function, I define a system \begin{eqnarray*} [ \ ] \colon OT \times \mathbb{N} & \to & OT \\ (a,n) & \mapsto & a[n] \end{eqnarray*} of fundamental sequences in the following way: I define a map \begin{eqnarray*} f \colon OT \times \mathbb{N} & \to & \mathbb{N} \\ (a,n) & \mapsto & f_a(n) \end{eqnarray*} in the following transfinitely inductive way: I denote by \begin{eqnarray*} F \colon \mathbb{N} \to \mathbb{N} \end{eqnarray*} the map which assigns \(f_{\alpha_n}(n)\) to each \(n \in \mathbb{N}\), where \(\alpha_n\) denote the \((1+n)\)-th element of \(OT\). I hope that the uncoumputable large number \(F^{10}(10)\) (with respect to a non-pathologic choice of an enumeration of Turing machines) is perhaps greater than any known computable large number. I have no idea about the comparison to the FGH on Kleene's \(\mathcal{O}\).
 * 1) If \(\textrm{ot}(a)\) is zero, then \(a[n] := a\).
 * 2) If \(\textrm{ot}(a)\) is a successor ordinal, then \(a[n] := \textrm{ot}^{-1}(\max \textrm{ot}(a))\).
 * 3) If \(\textrm{ot}(a)\) is a non-zero limit ordinal, then \(a[n] := \textrm{ot}^{-1}(n + \max (\{\beta \in \textrm{ot}(a) \mid \textrm{ot}^{-1}(\beta) < n\} \cup \{0\}))\).
 * 1) If \(\textrm{ot}(a)\) is zero, then \(f_a(n) := n+1\).
 * 2) If \(\textrm{ot}(a)\) is not zero, then \(\sum_{m=0}^{n} f_{a[m]}^m(m)\).