Rayo's number is a large number named after Mexican philosophy professor AgustÃÂn Rayo. It was claimed to be the largest named number when it was made. It was originally defined in a "Big Number Duel" at the Massachusetts Institute of Technology (MIT) on 26 January 2007.
The definition of Rayo's number is a variation on the definition:
<blockquote>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.</blockquote>
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".
The formal definition of the number defines a predicate according to the following second-order formula, where is a Gödel-coded formula and is a variable assignment:
Given this formula, Rayo's number is defined as:
<blockquote>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 .</blockquote>
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 instead of .
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 , and that is "Rayo-nameable" in symbols. Then, is defined as the smallest greater than all numbers Rayo-nameable in at most symbols.
Rayo notes that mathematicians working with a nonrealist philosophy of math may reject Rayo's number as being well-defined, because any axiomatization of the language of second-order logic will have non-isomorphic models, under which Rayo's number could correspond to different values.