Rayo's number
Rayo's number is a large number named after Mexican philosophy professor Agustín Rayo. It has been claimed to be the largest named number. It was originally defined in a "big number duel" at the Massachusetts [Institute of Technology] on 26 January 2007.
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 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:
The smallest number bigger than every finite number with the following property: there is a formula in the language of first-order set-theory with less than a googol symbols and as its only free variable such that: there is a variable assignment assigning to such that, and for any variable assignment, if, then assigns to.
Explanation
Intuitively, Rayo's number is defined in a formal language, such that:- and are atomic formulas.
- If is a formula, then is a formula.
- If and are formulas, then is a formula.
- If is a formula, then is a formula.
It is possible to express the missing logical connectives in this language. For instance:
- Disjunction: as.
- Implication: as.
- Biconditional: as.
- Universal quantification: as.