Nombre de Rayo

Le nombre de Rayo est l'un des plus grands nombres nommés, inventé lors d'une bataille de grands nombres appelée "Big Number Duel" opposant Agustín Rayo à Adam Elga. Le nombre de Rayo est, selon les propres termes de Rayo, "le plus petit nombre entier positif plus grand que tout nombre entier positif fini nommé par une expression dans le langage de la théorie des ensembles du premier ordre avec des symboles googol ou moins".

By letting the number of symbols range over the natural numbers, we get a very quickly growing function Rayo(n). Rayo's number is Rayo(10)¹⁰⁰. Rayo's function is uncomputable, which means that it is impossible for a Turing machine (and, by the Church–Turing thesis, any modern computer) to calculate Rayo(n). Indeed, if f is a function definable in a first order segment of the second order set theory assumed in the definition of Rayo's function, the defining formula Φ(n,m) of the predicate m = f(n) gives a lower bound of the composition of Rayo(n) and a sufficiently slow growing function depending on the growth rate of the length of $\exists m(\Phi(\ulcorner n \urcorner,m))$ regarded as a function on n, where $\ulcorner n \urcorner$ is a fixed formalisation of n in the language of first order set theory.

Although the second order set theory was unspecified in the original definition and is clarified as the philosophic (but mathematically ill-defined) collection of formulae which the real world philosophically "satisfy", it is reasonable to assume that ZFC set theory is a first order segment of the unspecified set theory because the majority of mathematicians and googologists are interested in ZFC set theory. Under the assumption, Rayo's function outgrows all functions definable in ZFC set theory. Throughout this article, we always use the same assumption except for Axiom section which more deeply explains the issue on the lack of the clarification of the second order set theory.

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-nameable 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-nameable in \(n\) symbols.

Note that the x_i's in t(x_i) ∈ t(x_j) and t(x_i) = t(x_j) in the definition of \(\text{Sat}\) above were x_1 in the original definition. Although x_1 is the only free variable allowed to occur in a Rayo-name, the variable assignment for x_i is actually referred to in the satisfaction of ∃-fourmulae. Therefore the original definition did not work as Rayo actually intended, and has been updated by Rayo himself. (Retrieved 19/05/2020)

Explanation 1

There are many terminologies of formal logic depending on authors. We explain one of such terminologies. A formal language is a set of constant term symbols, variable term symbols indexed by natural numbers, function symbols, and relation symbols. A formula in a formal language \(L\) is formal strings built from constant term symbols in \(L\), variable term symbols in \(L\), function symbols in \(L\), relation symbols in \(L\), quantifiers, and logical connectives following a certain syntax.

Formally adding every set as a constant term symbol, we can canonically extend \(L\) to a formal language \(L'\) with parameters in \(V\). An interpretation of formulae in \(L\) is canonically extended to an interpretation of formulae in \(L'\). A variable assignment is some countably infinite sequence of mathematical objects such as \(A = (3, 2, 6, 1/2, \{4, \pi\}, \omega, 65, \ldots)\). Given a variable assignment, we can convert a formula in \(L'\) into a closed formula in \(L'\) by replacing free occurrences of variable term symbols by the parameter given by the corresponding entry in the variable assignment.

For example, the formula "the third variable is relatively prime to the second variable" in \(L\) is interpreted by \(A\) as the closed formula "\(6\) is relatively prime to \(2\)" in \(L'\), and the formula "the second variable is the set of all real numbers except for \(3.2\)" in \(L'\) is interpreted by \(A\) as the closed formula "\(2\) is the set of all real numbers except for \(3.2\)" in \(L'\).

An interpretation of formulae in \(L\) is a map which assigns a constant to each constant term symbol, a function to each function symbol, and a relation to each relation symbol. Given an interpretation of formulae in \(L\), every closed formula in \(L'\) will be evaluated as true or false as long as a truth predicate is formalisable, because it corresponds to a formula on parameters in \(V\). In particular, given a variable assignment and an interpretation, we can ask whether a given formula in \(L\) is true or false as long as a truth predicate is formalisable. In order to formalise a truth predicate, we need a sufficiently strong set theory. For example, ZFC set theory is not suitable for this purpose.

Rayo defined a very specific and abstract formal language together with a canonical choice of an interpretation:

• An atomic formula "x_a∈x_b" means that the ath variable is an element of bth variable.
• An atomic formula "x_a=x_b" means the ath variable is equal to the bth variable.
• A formula "(¬e)" for a formula e means the negation of e.
• A formula "(e∧f)" for formulas e and f means the conjunction (the logical and) of e and f.
• A formula "∃x_a(e)" means that we can modify the free occurrence of the ath variable, i.e. substitute x_a by another member of the class V (von Neumann universe) of all sets, in e so that the formula e is true.

An atomic formula is a special sort of a formula.

For example, take the formula "(x₁∈x₂)". This says "the first variable is an element of the second variable," so we can plug in the variable assignment \((\emptyset, \{\emptyset\}, \ldots)\) and the result will be true since \(\emptyset\) is an element of \(\{\emptyset\}\). If instead we plug in \((\{\emptyset\}, \emptyset, \ldots)\), the result is not true because \(\{\emptyset\}\) is not an element of \(\emptyset\). (If you're familiar with propositional logic, you might be curious about the absence of the universal quantifier ∀. This is because ∀x(e) is the same as (¬(∃x((¬e)))), and it's not needed.)

A more complicated example: "(¬(∃x₁(x₁∈x₂)))". This says, "it is not true that there exists an x such that "the first variable is an element of the second variable" is not true when we replace "the first variable" by x. In other words, we cannot pick an x such that x is an element of the second variable. Given a variable assignment, this is true when the second entry is the empty set. For example, this is true for the variable assignment \((3, \{\}, \ldots)\), but is false for the variable assignment \((3,\{1\},\ldots)\).

If a formula returns true when a variable assignment is plugged into it, we say that the variable assignment "satisfies" that formula.

Now we arrive at the core concept of Rayo-nameability, ignoring the length restriction:

There is a formula \(\phi\) such that all satisfactory variable assignments must have \(m\) as their first argument, and there is at least one such assignment.

First we shall show that 0 is Rayo-nameable. In the ordinal system, \(0 = \{\}\). We need to craft a formula \(\phi\) that forces \(\{\}\) as its first argument. One such string is "(¬(∃x₂(x₂∈x₁)))" = "we cannot pick an \(x\) such that \(x\) is an element of the first variable" = "we cannot pick an element of the first variable" = "the first variable has no elements" = "the first variable is the empty set".

Now we need to find a way to Rayo-name \(x_1=\{\{\}\}\). The first variable needs an element: "∃x₂(x₂∈x₁)". Then, to ensure that the element is the empty set, we use "(￢(∃x₂((x₂∈x₁∧∃x₃(x₃∈x₂)))))" = "we cannot pick an \(x\) such that \(x\) is an element of the first variable and the third variable is an element of \(x\)" = "we cannot pick an \(x\) such that \(x\) is an element of the first variable and has an element" = "we cannot pick an \(x\) such that \(x\) is an element of the first variable and is not the empty set" = "if the first variable has an element, it must be the empty set". We "and" all these statements together, we get "(∃x₂(x₂∈x₁)∧(￢(∃x₂((x₂∈x₁∧∃x₃(x₃∈x₂))))))".

We can continue with this pattern, defining each natural number using this method. It allows us to name the number \(n\) in \(O(n^2)\) symbols. With larger values, it is possible to define recursive operations, allowing us to Rayo-name larger and larger numbers using compact notation. Given a sufficiently large number, a Rayo string that defines exponentiation would need less symbols than our naïve technique.

Note that the symbol x_n is counted as a single symbol - it should not be broken into separate symbols x and n.

We have all the pieces in place to define Rayo's function:

Rayo's function Rayo(n) is defined as the smallest non-negative integer greater than all non-negative integers Rayo-nameable in at most n symbols.

Why is Rayo's function uncomputable? Using Rayo's microlanguage one can construct a set whose elements 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 analogs of Busy Beaver function, which Googology Wiki user EmK has done.

Example Rayo strings and their values

\begin{eqnarray*} 0 & = & \{\} = (\neg\exists x_2(x_2 \in x_1)) \ \textrm{(10 symbols)} \\ 1 & = & \{\{\}\} = (\exists x_2(x_2 \in x_1) \land (\neg\exists x_2((x_2 \in x_1 \land \exists x_3(x_3 \in x_2))))) \ \textrm{(30 symbols)} \\ 2 & = & \{\{\},\{\{\}\}\} \\ & = & (\exists x_2(\exists x_3((x_3 \in x_2 \land (x_2 \in x_1 \land x_3 \in x_1)))) \\ & & \land (\neg \exists x_4(\exists x_2(\exists x_3((x_4 \in x_3 \land (x_3 \in x_2 \land x_2 \in x_1))))))) \ \textrm{(56 symbols)} \end{eqnarray*}

Thus:

\begin{eqnarray*} \text{Rayo}(0) &=& 0 \\ \text{Rayo}(10) &\ge& 1 \\ \text{Rayo}(30) &\ge& 2 \\ \text{Rayo}(56) &\ge& 3 \\ \end{eqnarray*}

Although this argument just gives lower bounds, the precise values for small values are given by Googology Wiki users Plain'N'Simple and Emk:

\begin{eqnarray*} \text{Rayo}(0) &=& 0 \\ \text{Rayo}(1) &=& 0 \\ &\vdots& \\ \text{Rayo}(9) &=& 0 \\ \text{Rayo}(10) &=& 1 \\ \text{Rayo}(11) &=& 1 \\ &\vdots& \\ \text{Rayo}(19) &=& 1 \\ \end{eqnarray*}

The Rayo function barely has square root growth rate for small values, but if we add a ton of symbols, we can represent much larger numbers.

Additionally, Plain'N'Simple has shown that \(\text{Rayo}(835+96n) > 2 \uparrow \uparrow n\). The bound was later reduced to Rayo(728+75n) > 2 ↑↑ n by an idea from Zongshu Wu, and to \(\text{Rayo}(731+65n) > 2↑↑n\) by an idea from P進大好きbot. P進大好きbot has shown that Rayo(816+42n) > 2↑↑n\), which is a better bound for \(n\ge 4\).

So:

\begin{eqnarray*} \text{Rayo}(861) &>& 4 \\ \text{Rayo}(926) &>& 16 \\ \text{Rayo}(984) &>& 65536 \\ \text{Rayo}(1026) &>& 2^{65536} \\ \end{eqnarray*}

Explanation 2

We will open with Berry's paradox:

Let x be the smallest natural number greater than all ones definable in at most fifteen English words. Then x can be defined as "the smallest natural number greater than all ones definable in at most fifteen English words." We just defined x using at most fifteen English words, so therefore x cannot be greater than all natural numbers definable in at most fifteen 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 first-order logic with the von Neumann universe 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 Tarski's undefinability theorem, which says we can't formally define truth, let alone definability, so FOST can't invoke FOST the way English can invoke English.

Axiom

In order to define a natural number using set theory, we need to fix under what axioms we define it. A problem in the definition of Rayo's number is that Rayo did not clarify the axioms. In mathematics, we traditionally omit the declaration of the axioms in which we are working as long as we are working in \(\textrm{ZFC}\) set theory. By the tradition, several googologists think that Rayo's number is defined in \(\textrm{ZFC}\) set theory, or is irrelevant to axioms, but it is wrong.

At least, since \(\textrm{ZFC}\) set theory is not able to formalize truth predicate at von Neumann universe, Rayo's number is ill-defined in \(\textrm{ZFC}\) set theory unless we interpret the definition of Rayo's number in terms of provability. Even if we interpret the definition in that way, the resulting large number will not be significantly larger than, for example, \(\Sigma(10^{100})\) (where \(\Sigma\) is the busy beaver function) because the provability in a recursively enumerable theory with a restriction of the length is decidable by a Turing machine. To get significantly beyond the Busy Beaver function, we must abandon provability, and talk about truth in a particular model, whose existence is not provable under \(\textrm{ZFC}\) set theory as long as \(\textrm{ZFC}\) set theory is consistent.

On the other hand, FOST is just a formal language, which is irrelevant to axioms by the definition, but it does not mean Rayo's number is irrelevant to axioms. The irrelevance of FOST and axioms or the relation between the Busy Beaver function and the proof-based interpretation of the definition of Rayo's number might be the main reasons of the misunderstanding that Rayo's number is irrelevant to axioms.

As Rayo wrote that he uses second-order set theory in order to formalize the primitive semantics vocabulary in the original description, Rayo's number is defined under certain axioms of second-order set theory, which are not clarified. It is important to clarify axioms in uncomputable googology, because uncomputable large numbers can be compared with each other only when they share axioms used in their definitions. Fortunately, there are many choices of axioms of second-order set theory which enable us to define Rayo's number. As a conclusion, Rayo's number is well-defined for googologists who do not care about clarification of axioms and is ill-defined for googologists who care about clarification of axioms.

In 2020, Rayo added the following new description of the way to deal with Rayo's number:

Note: Philosophers sometimes assume a realist interpretation of set-theory. On this interpretation,set-theoretic expressions have "standard" meanings, which determine a definite truth-value for each sentence of the language, regardless of whether it is in principle possible to know what those truth-values are. (See, for instance, this article by Vann McGee.) During the competition, Adam and I took for granted that the language of (second-order) set-theory was interpreted standardly, which guarantees that the final entry corresponds to a definite number. If the language had instead been interpreted on the basis of an axiom system, the final entry would have been invalid. This is because every (consistent) axiomatization of the language has non-isomorphic models and there is no guarantee that the final entry will correspond to the same number with respect to different models.

This means that Rayo considers a philosophical "interpretation" of set theoretic formulae with respect to the "truth" in the real world, which is unformalisable in mathematics, and does not intend a specific choice of axioms. It is one of reasonable directions of googology outside mathematics. On the other hand, the issue in the last sentence looks like an excuse about why they take the unformalisable "truth" for granted, but it does not make sense because the dependency of the value of a given number on a model is irrelevant to the "invalidity". In mathematics, there are many well-defined notions which are not absolute, i.e. depend on a model, e.g. the unique natural number \(n\) satisfying \((\text{CH} \to n=0) \land (\neg \text{CH} \to n=1)\). In googology, there are many large numbers which depend on a model, e.g. values of Busy Beaver function and especially S(1919), where S denotes maximum Shift function. In Big Number Duel, there is no rule prohibiting a number which depends on a model, and indeed it even allowed to skip to fix axioms.

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 Ph.D. in 2001.