Rayo's number

Rayo's number is a large number named after Mexican philosophy professor Agustín Rayo, who defined it during a "Big Number Duel" at the Massachusetts Institute of Technology (MIT) on 26 January 2007. At the time, it was claimed to be the largest named number, as its size vastly exceeds even other notable large numbers such as Graham's number and TREE(3).

Definition

The definition of Rayo's number is a variation on the definition:

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 (10100) symbols".

The formal definition of the number defines a predicate

Sat

{\displaystyle {\mbox{Sat}}}

according to the following second-order formula, where

[

ϕ

]

{\displaystyle [\phi ]}

is a Gödel-coded formula and

s

{\displaystyle s}

is a variable assignment:

Sat

(

[

ϕ

]

,

s

)

:=

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

)

}

{\displaystyle {\begin{aligned}{\mbox{Sat}}([\phi ],s):=&{\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}}}

Given this formula, Rayo's number is defined as:

The smallest number bigger than every finite number

m

{\displaystyle m}

with the following property: there is a formula

ϕ

(

x

1

)

{\displaystyle \phi (x_{1})}

in the language of first-order set-theory (as presented in the definition of

Sat

{\displaystyle {\mbox{Sat}}}

) with less than a googol symbols and

x

1

{\displaystyle x_{1}}

as its only free variable such that: (a) there is a variable assignment

s

{\displaystyle s}

assigning

m

{\displaystyle m}

to

x

1

{\displaystyle x_{1}}

such that

Sat

(

[

ϕ

(

x

1

)

]

,

s

)

{\displaystyle {\mbox{Sat}}([\phi (x_{1})],s)}

, and (b) for any variable assignment

t

{\displaystyle t}

, if

Sat

(

[

ϕ

(

x

1

)

]

,

t

)

{\displaystyle {\mbox{Sat}}([\phi (x_{1})],t)}

, then

t

{\displaystyle t}

assigns

m

{\displaystyle m}

to

x

1

{\displaystyle x_{1}}

.

Explanation

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

x

i

∈

x

j

{\displaystyle x_{i}\in x_{j}}

and

x

i

=

x

j

{\displaystyle x_{i}=x_{j}}

are atomic formulas.

If

θ

{\displaystyle \theta }

is a formula, then

(

¬

θ

)

{\displaystyle (\neg \theta )}

is a formula (the negation of

θ

{\displaystyle \theta }

).

If

θ

{\displaystyle \theta }

and

ξ

{\displaystyle \xi }

are formulas, then

(

θ

∧

ξ

)

{\displaystyle (\theta \land \xi )}

is a formula (the conjunction of

θ

{\displaystyle \theta }

and

ξ

{\displaystyle \xi }

).

If

θ

{\displaystyle \theta }

is a formula, then

∃

x

i

(

θ

)

{\displaystyle \exists x_{i}(\theta )}

is a formula (existential quantification).

It is not allowed to eliminate parentheses.