Rayo's number

Definition

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 the language of set theory with 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 (10¹⁰⁰) symbols."

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

∀R {
{for any (coded) formula [ψ] and any variable assignment t
(R( [ψ],t) ↔
( ([ψ] = `x_i ∈ x_j' ∧ t(x_1) ∈ t(x_j)) ∨
([ψ] = `x_i = x_j' ∧ t(x_1) = t(x_j)) ∨
([ψ] = `(∼θ)' ∧ ∼R([θ],t)) ∨
([ψ] = `(θ∧ξ)' ∧ R([θ],t) ∧ R([ξ],t)) ∨
([ψ] = `∃x_i (θ)' and, for some an xi-variant t' of t, R([θ],t'))
)} →
R([φ],s)}

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 φ(x₁) in the language of first-order set-theory (as presented in the definition of `Sat') with less than a googol symbols and x₁ as its only free variable such that: (a) there is a variable assignment s assigning m to x₁ such that Sat([φ(x₁)],s), and (b) for any variable assignment t, if Sat([φ(x₁)],t), then t assigns m to x₁.