Rayo's number

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.

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.

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

• "x_i∈x_j" and "x_i=x_j" are atomic formulas.
• If θ is a formula, then "(~θ)" is a formula (the negation of θ).
• If θ and ξ are formulas, then "(θ∧ξ)" is a formula (the conjunction of θ and ξ).
• If θ is a formula, then "∃x_i(θ)" is a formula (existential quantification).

Notice that it is not allowed to eliminate parentheses. For instance, one must write "∃x_i((~θ))" instead of "∃x_i(~θ)".

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

• Disjunction: "(θ∨ξ)" as "(~((~θ)∧(~ξ)))".
• Implication: "(θ⇒ξ)" as "(~(θ∧(~ξ)))".
• Biconditional: "(θ⇔ξ)" as "((~(θ∧ξ))∧(~((~θ)∧(~ξ))))".
• Universal quantification: "∀x_i(θ)" as "(~∃x_i((~θ)))".

The definition concerns formulas in this language that have only one free variable, specifically x₁. If a formula with length n is satisfied iff x₁ 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, Rayo(n) is defined as the smallest k greater than all numbers Rayo-nameable in at most n symbols.

To Rayo-name 0, which is the empty set, one can write "(¬∃x₂(x₂∈x₁))", which has 10 symbols. It can be shown that this is the optimal Rayo string for 0. Similarly, (∃x₂(x₂∈x₁)∧(¬∃x₂((x₂∈x₁∧∃x₃(x₃∈x₂))))), which has 30 symbols, is the optimal string for 1. Therefore, Rayo(n)=0 for 0≤n<10, and Rayo(n)=1 for 10≤n<30.

Additionally, it can be shown that Rayo(34+20n)>n and Rayo(260+20n)>ⁿ2.