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 () symbols". The formal definition of the number defines a predicate \mbox{Sat} according to the following
second-order formula, where [\phi] is a
Gödel-coded formula and s is a variable assignment: : \begin{align} \mbox{Sat}([\phi],s):= & \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 ==