Another way of generating ordinals
Let's say you want a method of generating really large (recursive) ordinals. There are three main ways to do that:
- create a function that takes in ordinals and returns ordinals (ordinal function, which is almost always an OCF),
- define a set of strings/treees/some other structure, define a subset of that set (a set of "standard expressions" S), establish an ordering < between standard expressions, prove that < is a well-oprdering, and generate ordinals by taking the order type of (S,
An OCF that doesn't get stuck
edit: this was a failure
almost
Let \(M\) be the least weakly Mahlo cardinal, \(\text{Ord}\) be the class of all ordinals, \(\mathcal{P}(\text{Ord})\) be the class of subsets of \(\text{Ord}\), and \(\text{Reg}\) be the class of all regular ordinals. Furthermore, let \(\mathbb{Z}^{+}\) be the class of all positive integers.
Define the mapping \(C:\text{Ord}^2\mapsto \mathcal{P}(\text{Ord})\), the mapping \(\delta:\text{Ord}\mapsto \mathcal{P}(\text{Ord})\), the mapping \(\chi:\text{Ord}\mapsto \text{Ord}\) and the mapping \(\psi:\text{Ord}\times\text{Reg}\mapsto\text{Ord}\) as follows:
\(C_0(\alpha,\beta) = \beta\cup\delta(\alpha)\cup\{0,M\}\)
\(C_{n+1}(\alpha,\beta) (n\in\mathbb{Z}^{+}) = C_n(\alpha,\beta)\cup\{\gamma+\epsilon,\omega_{\gamma},…
Question about the weakly Mahlo OCF
(The OCF in question is this one.)
Take \(\psi_{\chi(M)}\). Up to \(\alpha = M\), like the OP pointed out, we get erratic behavior, and we get \(\psi_{\chi(M)}(M) =\) the first fixed point \(\alpha\mapsto\chi(\alpha)\). However, after \(\alpha = M\), we still get erratic behavior: the \(\chi(\alpha)\) function gets stuck from \(\alpha = \psi_{\chi(M)}(M)\) to \(\alpha = M\), since we don't get \(\chi(M)\) and thus \(\psi_{\chi(M)}(\alpha)\) for \(\alpha\geq M\) in \(C(\alpha,\beta)\), which means that we can only build up on lower values of \(\psi_{\chi(M)}(\alpha)\) using addition, \(\omega^{\gamma}\) and \(\Omega_{\gamma}\). This gets us that \(\psi_{\chi(M)}(M+\alpha) =\) the \(\alpha\)th fixed point \(\beta\mapsto\Omega_{\beta}\) after …