Rayo's number

Rayo's number is one of the largest named numbers, coined in a large number battle pitting Agustín Rayo against Adam Elga. Rayo's number is, in Rayo's own words, "the smallest positive integer bigger than any finite positive integer named by an expression in the language of first order set theory with a googol symbols or less."

By replacing "googol" with any positive integer, we get a very quickly growing function \(\text{Rayo}(n)\) (sometimes denoted \(\text{FOST}(n)\) ). Rayo's number is \(\text{Rayo}(10^{100})\). Rayo's function is uncomputable, which means that it is impossible for a (and, by the Church-Turing thesis, any modern computer) to calculate \(\text{Rayo}(n)\). But order-\(n\) Turing machines are able to compute this function. Rayo's function is one of the fastest growing functions ever to arise in professional mathematics; only a few functions, especially its derivation, the FOOT (first-order oodle theory) function surpass it.

Definition
Let \([\phi]\) and \([\psi]\) be Gödel-coded formulas and \(s\) and \(t\) be variable assignments. Define \(\text{Sat}([\phi], s)\) as follows:

∀R { {    ∀[ψ], s: R([ψ],t) ↔ ([ψ] = "xᵢ ∈ xⱼ" ∧ t(xᵢ) ∈ t(xⱼ)) ∨ ([ψ] = "xᵢ = xⱼ" ∧ t(xᵢ) = t(xⱼ)) ∨ ([ψ] = "(¬θ)"   ∧ ¬R([θ], t)) ∨ ([ψ] = "([θ]∧ξ)" ∧ R([θ], t) ∧ R([ξ], t)) ∨ ([ψ] = "∃xᵢ(θ)" ∧ ∃t′: R([θ], t′)) (where t′ is a copy of t with xᵢ changed) } ⇒ R([ϕ],s) }

Call a natural number \(m\) "Rayo-namable in \(n\) symbols" if there is a formula \(\phi(x_1)\) with less than \(n\) symbols and \(x_1\) as its only free variable that satisfies the following properties:


 * 1) There is a variable assignment \(s\), assigning \(x_1 := m\), such that \(\text{Sat}([\phi(x_1)], s)\).
 * 2) For any variable assignment \(t\), if \(\text{Sat}([\phi(x_1)], t)\), \(t\) must have \(x_1 = m\).

\(\text{Rayo}(n)\), then, is the smallest number greater than all numbers Rayo-namable in \(n\) symbols.

Explanation 1
Disclaimer: Some background in set theory and mathematical logic is extremely helpful in understanding Rayo's number.

A variable assignment is some infinite sequence of objects such as \((3, 2, 6, 1/2, \{4, \pi\}, \text{Canada}, \of which are so called instantaneous descriptions of a Turing machine, and from this it's just a small step to define Busy Beaver function. With more effort, one can even construct oracle Turing machines and define their analogues of Busy Beaver function.

Values

 * 0 = {} = (¬∃a(a∈b))
 * 1 = \(\lbrace\lbrace\rbrace\rbrace\) = (¬∃a(a∈b))∧b∈c)∧(¬∃a(a∈c∧(¬a=b))
 * 2 = \(\lbrace\lbrace\rbrace,\lbrace\lbrace\rbrace\rbrace\rbrace\) = ((((¬∃a(a∈b))∧b∈c)∧(¬∃a(a∈c∧(¬a=b)))∧b∈d)∧c∈d)∧(¬∃a((a∈d∧(¬a=b))∧(¬a=c))

Explanation 2
We will open with :


 * Let x be the smallest natural number greater than all natural numbers definable in at most sixteen English words. Then x can be defined as "the smallest natural number greater than all natural numbers definable in at most sixteen English words." We just defined x using at most sixteen English words, so therefore x cannot be greater than all natural numbers definable in at most sixteen English words. This is a contradiction.

The source of the paradox is the ambiguity of the word "definable", and more fundamentally the ambiguity of the English language itself. Rayo's function circumvents these mathematical sins by replacing English with the language called first-order set theory (FOST). FOST is the language of with the  as the domain. Specifically, FOST is capable of determining set membership, quantifying over the universe, and applying logical operators. The nitty-gritty details of how this works are given above.

We fix the loophole that causes Berry's paradox, resulting in the following definition of Rayo(n), which is:


 * the smallest natural number greater than all natural numbers that can be uniquely identified by a FOST expression of at most n symbols

The paradox is gone now because the definability has been replaced with a formal language. FOST is subject to, which says we can't formally define truth, let alone definability, so FOST can't invoke FOST the way English can invoke English.

Unlike first-order logic, FOST doesn't include axioms. If we include axioms in FOST, Rayo's number will not be significantly larger than Busy Beaver function because it can only define numbers defined with provable functions in a recursively enumerable theory. To get significantly beyond the Busy Beaver function, we must abandon provability, and talk about truth in a particular model.

History
Rayo and Elga's number duel was inspired by the large number competitions described in the article "Who Can Name the Bigger Number?" by Scott Aaronson.

In January 2013, Adam Goucher claimed that \(\text{Rayo}(n)\) grows slower than his xi function. However, the claim turned out to be incorrect, because Goucher misunderstood the definition of Rayo's function as the "largest integer expressible uniquely by n symbols in first-order arithmetic (the language of Peano arithmetic).". The Second-order arithmetic is much stronger, and first order set theory is even stronger than that. First-order arithmetic's domain of discourse is the natural numbers, but first order set theory's domain of discourse is defined to be sets of the entire von Neumann universe. In fact, it can be shown that Rayo's function is much more powerful than the xi function.

Rayo's number was honored as the largest named number until 2014 when BIG FOOT was defined, using a non-naive extension of n-th order set theory, the first-order oodle theory. Note that naive extensions like \(\text{FOST}^{100}(10^{100})\) where recursion/iteration notation is used are not honored as breaking the record of Rayo's number. Although Rayo's number was previously surpassed in 2013 by Fish number 7, it is debatable whether that number is a good enough extension to be honored as breaking the record. Currently, all the largest named numbers share the same concept of Rayo's function, and all the non-naive extensions such as the FOOT function.

Author
The number was invented by Dr. Agustín Rayo, an Associate Professor of Linguistics and Philosophy at the Massachusetts Institute of Technology where he received his PhD in 2001.