Rayo's number is a
large number named after Mexican philosophy professor
AgustÃn Rayo which has been claimed to be the largest named number. It was originally defined in a "big number duel" at
MIT 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 the language of first-order set theory with a googol
A googol is the large number 10100. In decimal notation, it is written as the digit 1 followed by one hundred zeroes: 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, ...
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 (10
100) symbols."
The formal definition of the number uses the following
second-order
Second-order may refer to:
Mathematics
* Second order approximation, an approximation that includes quadratic terms
* Second-order arithmetic, an axiomatization allowing quantification of sets of numbers
* Second-order differential equation, a di ...
formula, where
�is a
Gödel-coded formula and s is a variable assignment:
For all R
Given this formula, Rayo's number is defined as:
The smallest number bigger than every finite number m with the following property: there is a formula φ(x1) in the language of first-order set-theory (as presented in the definition of ''Sat'') with less than a googol symbols and x1 as its only free variable such that: (a) there is a variable assignment s assigning m to x1 such that Sat( �(x1)s), and (b) for any variable assignment t, if Sat( �(x1)t), then t assigns m to x1.
Explanation
Intuitively, Rayo's number is defined in a
formal language, such that:
* "x
i∈x
j" and "x
i=x
j" are atomic formulas.
* If θ is a formula, then "(~θ)" is a formula (the
negation
In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...
of θ).
* If θ and ξ are formulas, then "(θ∧ξ)" is a formula (the
conjunction of θ and ξ).
* If θ is a formula, then "∃x
i(θ)" is a formula (
existential quantification).
Notice that it is not allowed to eliminate parenthesis. For instance, one must write "∃x
i((~θ))" instead of "∃x
i(~θ)".
It is possible to express the missing
logical connectives
In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binar ...
in this language. For instance:
*
Disjunction: "(θ∨ξ)" as "(~((~θ)∧(~ξ)))".
*
Implication: "(θ⇒ξ)" as "(~(θ∧(~ξ)))".
*
Biconditional: "(θ⇔ξ)" as "((~(θ∧ξ))∧(~((~θ)∧(~ξ))))".
*
Universal quantification: "∀x
i(θ)" as "(~∃x
i((~θ)))".
The definition concerns formulas in this language that have only one
free variable, specifically x
1. If a formula with length n is satisfied
iff
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
x
1 is equal to the finite
von Neumann ordinal k, we say such a formula is a "Rayo string" for k, and that k is "Rayo-nameable" in n symbols. Then, Rayo(n) is defined as the smallest k greater than all numbers Rayo-nameable in at most n symbols.
Examples
To Rayo-name 0, which is the empty set, one can write "(¬∃x
2(x
2∈x
1))", which has 10 symbols. It can be shown that this is the optimal Rayo string for 0. Similarly, (∃x
2(x
2∈x
1)∧(¬∃x
2((x
2∈x
1∧∃x
3(x
3∈x
2))))), which has 30 symbols, is the optimal string for 1. Therefore, Rayo(n)=0 for 0≤n<10, and Rayo(n)=1 for 10≤n<30.
Additionally, it can be shown that Rayo(34+20n)>n and Rayo(260+20n)>
n2.
References
{{Use dmy dates, date=February 2018
Large integers