Eli5- what exactly is Rayo’s number?
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It is a very large (arguably the largest finite number eveer) number that came out of a competition trying to produce the largest finite number possible. The competition had a few important rules namely the number can’t be arbitrary(your number plus 1) and cant be a trivial extension of a smaller number(TREE(4)>TREE(3)). Rayo’s number in plain English is:
>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
To understand what this mean you first need to know the basics of first order set theory. Basically FOST is an attempt to start with the most basic symbols of logic and build every thing up from there. For small numbers this is very inefficient, for example to define in ‘0’ FOST you need to write:
>∃x1¬∃x2(x2∈x1)
and to define 1 you have to write:
>∀x.x∈1↔(∀y.y∉x)
and to define 2 you have to write:
>∃x1∀x2(x2∈x1↔(¬∃x3(x3∈x2)∨∀x3(x3∈x2↔¬∃x4(x4∈x3))))
As you can see this starts out very inefficient but eventually we can start to build functions. Functions are a much more efficient way to write big numbers. As an example consider writing a google with just numbers, that would take 101 symbols (1 followed by 100 0’s). But with a function we can just write 10^100 once we have defined what the function x^y means.
This is basically what Rayo did, he came up with a way to define a very efficient function that is still finite but very large.
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