Rayo's number

Rayo's number is a large number named after Mexican associate professor  [es] (born 1973) which has been claimed to be the largest (named) number.^[1][2] It was originally defined in a "big number duel" at MIT on 26 January 2007.^[3][4]

Definition[]

The definition of Rayo's number is a variation on the definition:^[5]

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

${\displaystyle Rayo(10^{100})BC}$

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."^[4]

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:^[5]

For all R {
{for any (coded) formula [ψ] and any variable assignment t
(R([ψ],t) ↔
(([ψ] = "x_i ∈ x_j" ∧ t(x_i) ∈ t(x_j)) ∨
([ψ] = "x_i = x_j" ∧ t(x_i) = 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 x_i-variant t' of t, R([θ],t'))
)}   →
R([φ],s)}

Given this formula, Rayo's number is defined as:^[5]

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.

References[]