User blog:P進大好きbot/Examples of Rules of Large Number Contests

As I wrote in my previous blog post, Rayo's number (and any other uncomputable large numbers based on similar systems) is not well-defined even if we assume the abstract existence of a Platonist universe. The point was that there is no meta-theoretic way without paradoxes to refer to the truth in it.

Instead, I consider explicit rules on large number contests, which allow us to deal with well-defned Rayo-like large numbers.

Let \(A\) denote the ZFC axiom or any other explicit recursive axiom of set theory that you like in the following.

= Model Rule =

Define a function formula \(F\) in \(A\) assigning to each model \(M\) of \(A\) a natural number in \(M\).

More precisely, \(F\) is a formula such that \begin{eqnarray*} A \vdash (\forall M, \forall E,((M,E) \models A) \to (\exists ! n \in \omega^{(M,E)}, F)))), \end{eqnarray*} where \(\omega^{(M,E)}\) denotes the set of natural numbers relative to \(M\).

I give a precise description. Let \(L_V\) denote the formal language of coded set theory associated to the proper class \(V\). More precisely, every class in this context is interpreted to a formula in set theory in an elementary way.

I denote by \(\textrm{Namable} \subset \omega\) the subset consisting of such an \(m\) that there exists a formula \(\Phi\) in \(L_V\) of length smaller than \(10^{100}+1\), with a free occurence of (the variable canonically corresponds to) \(m\), and with no parameters in \(V\) for which \(m\) satisfies \(\Phi \wedge (\forall m'(\Phi[m'/m] \to (m'=m)))\) in the coded sense.

I note that since the satisfaction is appropriately coded, \(\textrm{Namable}\) is a definable subset of \(\omega\). Therefore the subset \(\textrm{Nameble}^{(M,E)} \subset^{(M,E)} \omega^{(M,E)}\) makes sense as an element of \(M\) by the axiom of separation. By the definition of \(\omega^{(M,E)}\) given by the axiom of infinity relative to \((M,E)\), \(\omega^{(M,E)} \setminus^{(M,E)} \textrm{Namable}^{(M,E)}\) admits the minimum \(n\).

= Maximal Consistent Axiom Rule =

Define a map \(F\) in \(A\) assigning to each maximal consistent axiom \(\overline{A}\) in \(\textrm{FOST}\) containing the G\"odel-codes of axioms in \(A\) a natural number in \(\overline{A}\).

More precisely, \(F\) is a map in \(A\) from is the set \(\textrm{MCA}\) of maximal consistent axioms \(\overline{A}\) in \(\textrm{FOST}\) containing the G\"odel-codes of \(A\) to the set \(\textrm{Formula}_{textrm{FOST}}\) of formulae in \(\textrm{FOST}\) satisfying \begin{eqnarray*} A \vdash (\forall \overline{A} \in \textrm{MCA}, (\overline{A} \vdash (\exists ! n \in \omega, F(\overline{A})))), \end{eqnarray*} where \(F(\overline{A})\) is the abbreviation of the formula in \(\textrm{FOST}\) characterised by \((\overline{A},F(\overline{A}) \in  F)\).

I give a precise description. I define a map \(\textrm{code} \colon \omega\ \to \textrm{Formula}_{\textrm{FOST}}\) by the following classical way known as the von Neumann construction: \begin{eqnarray*} \textrm{code}(0) & := & \emptyset \\ \textrm{code}(n+1) & := & \textrm{code}(n) \cup \{\textrm{code}(n)\} \end{eqnarray*} You know that there are no symbols like \(\emptyset\), \(\cup\), and \(\{\}\) in \(\textrm{FOST}\), but it is not so harmful to apply the obvious abbreviation because we only consider axioms containing \(A\).

I denote by \(\textrm{Namable}_{\overline{A}} \subset \omega\) the subset consisting of such an \(m\) that there exists a formula \(\Phi\) in \(\textrm{FOST}\) of length smaller than \(10^{100}+1\) with a free occurence of a variable \(x\) for which \(\overlin{A}\) contains \(\Phi[\textrm{code}(m)/x] \wedge (\forall m'(\Phi[m'/x] \to (m'=\textrm{code}(m))))\) in the coded sense.

I note that \(\textrm{Namable}_{\overline{A}}\) is a real set, but is not a coded one. Therefore it is impossible for the coded set theory based on \(\overline{A}\) to refer to \(\textrm{Namable}_{\overline{A}}). At least, \(\omega \setminus \textrm{Namable}_{\overline{A}}\) admits the minimum \(n\) as a real set, and hence we can define \(F(\overline{A})\) as the defining formula \(\textrm{code}(n) \in \textrm{Formula}_{\textrm{FOST}\) in \(\overline{A}\).

It is remarkable that "the truth" of a formula \(\Phi\) in \(\textrm{FOST}\) is interpreted as the provability of \(\Phi\) by \(\overline{A}\), which is equivalent to the relation \(\Phi \in \overline{A}\) by the maximality of \(\overline{A}\). The provability is enough strong here, because \(\overline{A}\) is far from recursive.