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 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 (
"[4] The formal definition of the number uses the following second-order formula, where
[ ϕ ]
is a Gödel-coded formula and
is a variable assignment:[5]
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
, and (b) for any variable assignment
.Intuitively, Rayo's number is defined in a formal language, such that: Notice that it is not allowed to eliminate parentheses.
For instance, one must write
It is possible to express the missing logical connectives in this language.
For instance: 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
is defined as the smallest
greater than all numbers Rayo-nameable in at most