User:Vel!/Nonrecursive ordinals

The wacky world beyond \(\).

Kleene's \(\mathcal{O}\)

 * \(0 \in K\) and \(\mathcal{O}(0) = 0\).
 * If \(n \in K\) and \(\mathcal{O}(n) = \alpha\), then \(2^n \in K\), \(\mathcal{O}(2^n) = \alpha + 1\), and \(n <_\mathcal{O} 2^n\).


 * If for all natural numbers \(n\), we have \(f_i(n) \in K\) and \(f_i(n) <_\mathcal{O} f_i(n + 1)\), then \(3 \cdot 5^i \in K\), \(\mathcal{O}(3 \cdot 5^i) = \lim_{k \rightarrow \omega} \mathcal{O}(f_i(k))\), and for all \(k\) \(f_i(k) <_\mathcal{O} 3 \cdot 5^i\).
 * \(a <_\mathcal{O} b\) and \(b <_\mathcal{O} c\) implies \(a <_\mathcal{O} c\).

Define \(u(0) = 0\) and let \(u(n + 1)\) be the smallest \(m\) such that \(\mathcal{O}(m)\) is defined and \(m <_\mathcal{O} n\). \([n] = \mathcal{O}(u(n))\).

To a constant
Next we can introduce a variant of \(\mathcal{O}\) that gives it access to an ordinal \(\gamma\). Add the following rule:


 * \(7 \in K\) and \(\mathcal{O}(7) = \gamma\).

To a function
Let \(\gamma\) be a \(\Omega \mapsto \Omega\). Add this rule:


 * If \(n \in K\) and \(\mathcal{O}(n) = \alpha\), then \(5 \cdot 7^n \in K\), \(\mathcal{O}(5 \cdot 7^n) = \gamma(\alpha)\), and \(n <_\mathcal{O} 5 \cdot 7^n\).

This is a generalization of the previous function. You can pick \(\gamma\) so it has only 0 in its domain and let \(\gamma(0)\) be the constant.

We can construct a variant \(\mathcal{O}_1\) of Kleene's \(\mathcal{O}\) relative to \(\alpha \mapsto \omega^\text{CK}_\alpha\).