OFFSET

0,31

COMMENTS

The alphabet consists of the 7 symbols ∈, =, (, ), ~, ∧, ∃, plus infinitely many variables x_i, for i >= 0, each considered one symbol.

"x_i∈x_j" and "x_i=x_j" are atomic formulas.

If θ is a formula, then "(~θ)" is a formula (the negation 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).

A formula having x_0 as its only free variable defines a number m if it has a satisfying assignment, and all such assignments assign m to x_0.

Mexican philosophy professor Agustín Rayo defined a nondecreasing version of this function in a "big number duel" at MIT on 26 January 2007.

Its value at n=10^100 is known as Rayo's number.

This is a set theoretic analog of the busy beaver function, which it easily outgrows.

LINKS

Googology Fandom, Low-level Rayo bounds

Googology Fandom, Proof that Rayo(n) is 1 for n between 10 to 19

Wikipedia, Rayo's Number

EXAMPLE

a(30)=2, since the 30 symbol formula "(∃x_1(x_1∈x_0)∧(¬∃x_1(∃x_2((x_2∈x_1∧x_1∈x_0)))))" uniquely defines the number 1, while smaller formulae can only define the number 0, the smallest being the 10 symbol "(¬∃x_1(x_1∈x_0))".

CROSSREFS

KEYWORD

nonn,more

AUTHOR

John Tromp, Dec 07 2023

STATUS

approved