Rayo's number is a large number named after Mexican philosophy professor Agustín Rayo 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]
The Rayo function of a natural number 
, notated as 
, is the smallest number bigger than every finite number 
with the following property: there is a formula 
in the language of first-order set-theory (as presented in the definition of 
) with less than symbols and 
as its only free variable such that: (a) there is a variable assignment 
assigning to such that ![{\displaystyle {\mbox{Sat}}([\phi (x_{1})],s)}](//wikimedia.org/api/rest_v1/media/math/render/svg/13ab4fd131db14f91575a902fc5abf6995f3bd8c)
, and (b) for any variable assignment 
, if ![{\displaystyle {\mbox{Sat}}([\phi (x_{1})],t)}](//wikimedia.org/api/rest_v1/media/math/render/svg/d8392e6a89759c59d13861e3ca4af2165d982ae4)
, then assigns to . This definition is given by the original definition of 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 any language of first-order set theory in which the language uses only 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 (
) symbols."[4]
The formal definition of the number uses the following second-order formula, where ![{\displaystyle [\phi ]}](//wikimedia.org/api/rest_v1/media/math/render/svg/ea331bb6ec06f4426444af6f2cc1d5ed7665f6ae)
is a Gödel-coded formula and is a variable assignment:[5]
![{\displaystyle {\begin{aligned}&{\mbox{For all }}R\ \{\\&\{{\mbox{for any (coded) formula }}[\psi ]{\mbox{ and any variable assignment }}t\\&(R([\psi ],t)\leftrightarrow \\&(([\psi ]={\mbox{''}}x_{i}\in x_{j}{\mbox{''}}\land t(x_{i})\in t(x_{j}))\ \lor \\&([\psi ]={\mbox{''}}x_{i}=x_{j}{\mbox{''}}\land t(x_{i})=t(x_{j}))\ \lor \\&([\psi ]={\mbox{''}}(\neg \theta ){\mbox{''}}\land \neg R([\theta ],t))\ \lor \\&([\psi ]={\mbox{''}}(\theta \land \xi ){\mbox{''}}\land R([\theta ],t)\land R([\xi ],t))\ \lor \\&([\psi ]={\mbox{''}}\exists x_{i}\ (\theta ){\mbox{'' and, for some an }}x_{i}{\mbox{-variant }}t'{\mbox{ of }}t,R([\theta ],t'))\\&)\}\rightarrow \\&R([\phi ],s)\}\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/0b1540333251782558fcc8dfeed97cef5ac2b1d4)
Given this formula, Rayo's number is defined as:[5]
The smallest number bigger than every finite number with the following property: there is a formula in the language of first-order set-theory (as presented in the definition of ) with less than a googol symbols and as its only free variable such that: (a) there is a variable assignment assigning to such that , and (b) for any variable assignment , if , then assigns to .
Intuitively, Rayo's number is defined in a formal language, such that:
and 
are atomic formulas.- If
is a formula, then 
is a formula (the negation of ).
- If and
are formulas, then 
is a formula (the conjunction of and ).
- If is a formula, then
is a formula (existential quantification).
Notice that it is not allowed to eliminate parentheses. For instance, one must write 
instead of 
.
It is possible to express the missing logical connectives in this language. For instance:
- Disjunction:
as 
.
- Implication:
as 
.
- Biconditional:
as 
.
- Universal quantification:
as 
.
The definition concerns formulas in this language that have only one free variable, specifically . If a formula with length is satisfied iff is equal to the finite von Neumann ordinal 
, we say such a formula is a "Rayo string" for , and that is "Rayo-nameable" in symbols. Then, is defined as the smallest greater than all numbers Rayo-nameable in at most symbols.
![{\displaystyle {\mbox{Sat}}([\phi (x_{1})],s)}](http://wikimedia.org/api/rest_v1/media/math/render/svg/13ab4fd131db14f91575a902fc5abf6995f3bd8c)
![{\displaystyle {\mbox{Sat}}([\phi (x_{1})],t)}](http://wikimedia.org/api/rest_v1/media/math/render/svg/d8392e6a89759c59d13861e3ca4af2165d982ae4)
![{\displaystyle [\phi ]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/ea331bb6ec06f4426444af6f2cc1d5ed7665f6ae)
![{\displaystyle {\begin{aligned}&{\mbox{For all }}R\ \{\\&\{{\mbox{for any (coded) formula }}[\psi ]{\mbox{ and any variable assignment }}t\\&(R([\psi ],t)\leftrightarrow \\&(([\psi ]={\mbox{''}}x_{i}\in x_{j}{\mbox{''}}\land t(x_{i})\in t(x_{j}))\ \lor \\&([\psi ]={\mbox{''}}x_{i}=x_{j}{\mbox{''}}\land t(x_{i})=t(x_{j}))\ \lor \\&([\psi ]={\mbox{''}}(\neg \theta ){\mbox{''}}\land \neg R([\theta ],t))\ \lor \\&([\psi ]={\mbox{''}}(\theta \land \xi ){\mbox{''}}\land R([\theta ],t)\land R([\xi ],t))\ \lor \\&([\psi ]={\mbox{''}}\exists x_{i}\ (\theta ){\mbox{'' and, for some an }}x_{i}{\mbox{-variant }}t'{\mbox{ of }}t,R([\theta ],t'))\\&)\}\rightarrow \\&R([\phi ],s)\}\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/0b1540333251782558fcc8dfeed97cef5ac2b1d4)
