Rayo's number

Rayo's number is a large number named after Mexican philosophy professor Agustín Rayo which has been claimed to be the largest named number. It was originally defined in a "big number duel" at MIT on 26 January 2007.

Definition
The Rayo function of a natural number $$n$$, notated as $$\mbox{Rayo}(n)$$, is the smallest number bigger than every finite number $$m$$ with the following property: there is a formula $$\phi(x_1)$$ in the language of first-order set-theory (as presented in the definition of $$\mbox{Sat}$$) with less than $$n$$ symbols and $$x_1$$ as its only free variable such that: (a) there is a variable assignment $$s$$ assigning $$m$$ to $$x_1$$ such that $$\mbox{Sat}([\phi(x_1)],s)$$, and (b) for any variable assignment $$t$$, if $$\mbox{Sat}([\phi(x_1)],t)$$, then $$t$$ assigns $$m$$ to $$x_1$$. This definition is given by the original definition of Rayo's number.

The definition of Rayo's number is a variation on the definition:

"The smallest number bigger than any finite number named by an expression in any language of first-order set theory in which the language uses only a googol symbols or less."

Specifically, an initial version of the definition, which was later clarified, read "The smallest number bigger than any number that can be named by an expression in the language of first-order set-theory with less than a googol ($$10^{100}$$) symbols."

The formal definition of the number uses the following second-order formula, where $$[\phi]$$ is a Gödel-coded formula and $$s$$ is a variable assignment:


 * $$\begin{align}

& \mbox{For all }R \ \{ \\ & \{ \mbox{for any (coded) formula } [\psi] \mbox{ and any variable assignment } t \\ & (R([\psi],t) \leftrightarrow \\   & (( [\psi] = \mbox{}x_i \in x_j\mbox{} \and t(x_i) \in t(x_j)) \ \or \\ & ( [\psi] = \mbox{}x_i = x_j\mbox{} \and t(x_i) = t(x_j)) \ \or \\ & ( [\psi] = \mbox{}(\neg \theta)\mbox{} \and \neg R([\theta],t)) \ \or \\ & ( [\psi] = \mbox{}(\theta \and \xi)\mbox{} \and R([\theta],t) \and R([\xi],t)) \ \or \\ & ( [\psi] = \mbox{}\exists x_i \ (\theta)\mbox{ and, for some an } x_i \mbox{-variant } t' \mbox{ of } t, R([\theta],t')) \\ & )\} \rightarrow \\   & R([\phi],s)\} \end{align}$$

Given this formula, Rayo's number is defined as:

"The smallest number bigger than every finite number $m$ with the following property: there is a formula $\phi(x_1)$ in the language of first-order set-theory (as presented in the definition of $\mbox{Sat}$) with less than a googol symbols and $x_1$ as its only free variable such that: (a) there is a variable assignment $s$ assigning $m$ to $x_1$ such that $\mbox{Sat}([\phi(x_1)],s)$, and (b) for any variable assignment $t$, if $\mbox{Sat}([\phi(x_1)],t)$, then $t$ assigns $m$ to $x_1$."

Explanation
Intuitively, Rayo's number is defined in a formal language, such that:


 * $$\mbox{}x_i \in x_j\mbox{}$$ and $$\mbox{}x_i = x_j\mbox{}$$ are atomic formulas.
 * If $$\theta$$ is a formula, then $$\mbox{}(\neg\theta)\mbox{}$$ is a formula (the negation of $$\theta$$).
 * If $$\theta$$ and $$\xi$$ are formulas, then $$\mbox{}(\theta \and \xi)\mbox{}$$ is a formula (the conjunction of $$\theta$$ and $$\xi$$).
 * If $$\theta$$ is a formula, then $$\mbox{}\exists x_i(\theta)\mbox{}$$ is a formula (existential quantification).

Notice that it is not allowed to eliminate parentheses. For instance, one must write $$\mbox{}\exists x_i((\neg \theta))\mbox{}$$ instead of $$\mbox{}\exists x_i(\neg \theta)\mbox{}$$.

It is possible to express the missing logical connectives in this language. For instance:


 * Disjunction: $$\mbox{}(\theta \or \xi)\mbox{}$$ as $$\mbox{}(\neg((\neg \theta)\and(\neg \xi)))\mbox{}$$.
 * Implication: $$\mbox{}(\theta \Rightarrow \xi)\mbox{}$$ as $$\mbox{}(\neg(\theta \and(\neg \xi)))\mbox{}$$.
 * Biconditional: $$\mbox{}(\theta \Leftrightarrow \xi)\mbox{}$$ as $$\mbox{}((\neg(\theta \and \xi))\and(\neg((\neg \theta) \and (\neg \xi))))\mbox{}$$.
 * Universal quantification: $$\mbox{}\forall x_i(\theta)\mbox{}$$ as $$\mbox{}(\neg \exists x_i((\neg \theta)))\mbox{}$$.

The definition concerns formulas in this language that have only one free variable, specifically $$x_1$$. If a formula with length $$n$$ is satisfied iff $$x_1$$ is equal to the finite von Neumann ordinal $$k$$, we say such a formula is a "Rayo string" for $$k$$, and that $$k$$ is "Rayo-nameable" in $$n$$ symbols. Then, $$\mbox{Rayo}(n)$$ is defined as the smallest $$k$$ greater than all numbers Rayo-nameable in at most $$n$$ symbols.

Examples
To Rayo-name $$0$$, which is the empty set, one can write $$\mbox{}(\neg \exists x_2(x_2 \in x_1))\mbox{}$$, which has 10 symbols. It can be shown that this is the optimal Rayo string for $$0$$. Similarly, $$\mbox{}(\exists x_2(x_2 \in x_1)\and(\neg \exists x_2((x_2 \in x_1 \and \exists x_3(x_3 \in x_2)))))\mbox{}$$, which has 30 symbols, is the optimal string for $$1$$. Therefore, $$\mbox{Rayo}(n)=0$$ for $$0 \leq n<10$$, and $$\mbox{Rayo}(n)=1$$ for $$10 \leq n<30$$.

Additionally, it can be shown that $$\mbox{Rayo}(34+20n)>n$$ and $$\mbox{Rayo}(260+20n)> {^{n}2}$$ (tetration).