One amazing step for man…

This is a really big step.

RAYO’S NUMBER.

It’s defined as Rayo(10^100).

For all I know, it might end with 7.

What is Rayo(n)?

So you see, there used to be this battle at MIT named THE BIG NUMBER DUEL

The duel was not a fighting duel. No! This duel was to see which person could think of the largest number that wasn’t infinite. It had to keep ascending, and you couldn’t do the same thing as whatever happened before, and no adding zero or plus one. You can’t do “whatever the other person can think of +1”, that’s not allowed either.

It was Agustín Rayo versus Adam Elga. Rayo went first. He wrote down a series of ones, about thirty or forty of them. Elga went to the blackboard (yes, there is a blackboard), and for all ones except the first two, he turned them into !

11!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! is about… well, that’s huge. 11! is about 40 million, 11!! is already undefined to Google, and 11!!! can’t be written down in scientific notation. Then with about 30 factorials… that’s HUGE.

Then Rayo’s response was this: BB(10^100). Why is there a BB gun involved? It’s not. Busy Beavers are involved.

The Busy Beaver function involves Turing machines, and let’s say the machine has n states and a halt state, where n is a positive integer, and one of the n states is a starting state, so if the list of states was 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … then state 1 would be the starting state.

The machine uses a two-way infinite tape. I’m guessing that it only goes left and right, it can’t go to another tape or something. The tape “alphabet” is {0, 1}, so if the tape was a hotel, it would be like an infinite row of hotel rooms, and the 0 would be like lights off and 1 would be like lights on.

I’m not a computer scientist.

The machine’s “transition function” (wtf dude) takes two inputs: the current state that is not the halt state, and the symbol in the current cell, a 0 or a 1.

The transition function… you know what? I regret writing all this. This is a post for next time.

So, I’ll be writing about the Busy Beavers next time.

Elga wrote, on the blackboard, Super Busy Beaver(10^100). That’s also going to show up soon.

Rayo came up with this: Rayo(10^100).

Well, he came up with:

(now, this is directly copied)

For all R {
{for any (coded) formula [ψ] and any variable assignment t
(R([ψ],t) ↔
(([ψ] = “x_i ∈ x_j” ∧ t(x_i) ∈ t(x_j)) ∨
([ψ] = “x_i = x_j” ∧ t(x_i) = t(x_j)) ∨
([ψ] = “(∼θ)” ∧ ∼R([θ],t)) ∨
([ψ] = “(θ∧ξ)” ∧ R([θ],t) ∧ R([ξ],t)) ∨
([ψ] = “∃x_i (θ)” and, for some an x_i-variant t’ of t, R([θ],t’))
)}   →
R([φ],s)}

(Well, except in actual set theory language)

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([φ(x₁)],s), and (b) for any variable assignment t, if Sat([φ(x₁)],t), then t assigns m to x₁

Okay, what?

Rayo’s number is the smallest number larger than any finite number named by an expression in the language of first order set theory in less than or equal to a googol symbols.

You know what? I’ll do a detail post on this, including physics, with credit to Wikipedia and Numberphile. Again.