User blog:P進大好きbot/ZFC-variant of Rayo's number

This is an English translation of my Japanese article submitted to a Japanese googology event. The regulation of the event only accepts large numbers defined in \(\textrm{ZFC}\) set theory. Therefore large numbers which require stronger formal theory, e.g. Rayo's number, are invalid. Instead, I construct countably many models of finite segments of \(\textrm{ZFC}\) set theory, approximating \(V\) through reflection principle, and define a replica of Rayo's number using satisfaction at those models. Unlike Rayo's number, it does not require truth predicate at \(V\), it is well-defined in \(\textrm{ZFC}\) set theory.

= Definition =

Let \(L\) denote the formal language of \(\textrm{ZFC}\) set theory consisting of variable term symbols \(x_0,x_1,\ldots\) indexed by natural numbers, a \(2\)-ary relation symbol \(\in\), logical connectives \(\neg\) and \(\to\), a universal quantifier \(\forall\), and embraces \((\) and \\).

A pair \((S,f)\) of a finite set \(S\) of closed \(L\)-formulae and an \(L\)-formula \(f\) with free occurrence of precisely one variable symbol is said to be satisfiable at a segment of \(V\) if there exists a pair \((\alpha,n)\) of an ordinal \(\alpha\) and an \(n \in \mathbb{N}\) such that \(\mathbb{N} \subset V_{\alpha}\), \(V_{\alpha} \models S\), and \((V_{\alpha},n) \models f\). Here, \(V_{\alpha}\) denotes the cumulative hierarchy.

I denote by \(\Sigma\) the set of pairs satisfiable at a segment of \(V\). I define a map \begin{eqnarray*} \textrm{eval} \colon \Sigma & \to & \mathbb{N} \\ (S,f) & \mapsto & \textrm{eval}(S,f) \end{eqnarray*} in the following way:
 * 1) I denote by \(\alpha_0\) the least ordinal \(\alpha\) satisfying that there exists an \(n \in \mathbb{N}\) such that \(\mathbb{N} \subset V_{\alpha}\), \(V_{\alpha} \models S\), and \((V_{\alpha},n) \models f\).
 * 2) I define \(\textrm{eval}(S,f)\) as the least \(n \in \mathbb{N}\) satisfying \((V_{\alpha_0},n) \models f\).

For any \(n \in \mathbb{N}\), I define \(\textrm{CoRayo}(n) \in \mathbb{N}\) as the least \(m \in \mathbb{N}\) satisfying that for any \((S,f) \in \Sigma\), if all elements of \(S\) and \(f\) are of length \(\leq n \uparrow^n n\), then \(\textrm{eval}(S,f) < m\).

I guess that \(\textrm{CoRayo}\) is the greatest large function ever defined in \(\textrm{ZFC}\) set theory. For a stronger set theory, see another blog post. I guess that the large function defined there is the greatest in valid googology.