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Rayo's function: smallest natural number larger than any number uniquely defined by an n-symbol formula in First Order Set Theory.
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%I #21 Dec 19 2023 12:51:55

%S 0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,1,0,1,1,1,0,1,1,1,1,1,1,1,1,2

%N Rayo's function: smallest natural number larger than any number uniquely defined by an n-symbol formula in First Order Set Theory.

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

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

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

%C If θ and ξ are formulas, then "(θ∧ξ)" is a formula (the conjunction of θ and ξ).

%C If θ is a formula, then "∃x_i(θ)" is a formula (existential quantification).

%C 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.

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

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

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

%H Googology Fandom, <a href="https://googology.fandom.com/wiki/User_blog:12AbBa/Low-level_Rayo_bounds">Low-level Rayo bounds</a>

%H Googology Fandom, <a href="https://googology.fandom.com/wiki/User_blog:Plain&#39;N&#39;Simple/Proof_that_Rayo(n)_is_1_for_n_between_10_to_19">Proof that Rayo(n) is 1 for n between 10 to 19</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Rayo%27s_number">Rayo's Number</a>

%e 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))".

%Y Cf. A028444.

%K nonn,more

%O 0,31

%A _John Tromp_, Dec 07 2023