Eli5- what exactly is Rayo’s number?

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It’s the largest number known. Previously the largest number was Grahams number , which was summarised as, “if your wrote one number on every single particle in the universe, you would run out of particles before you even made a dent in the number”

Rayo’s number is bigger than Grahams number.

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.

There are infinitely many numbers. It’s hard to wrap your head around just how stupidly much “infinitely many” is. Hence, for the amusement of themselves and others, some mathematicians have taken up a pastime of pointing out humongously large numbers that demonstrate just how tiny the numbers we’re working with are compared to what’s available.

Rayo’s number comes from a competition where two contestants competed who can name the biggest number. To have a competition like that make sense, your answer would have to involve some new idea that makes it significantly bigger than the previous one: you shouldn’t be allowed to just say “previous answer times 100” or something. Rayo’s number was the biggest number that came out of that competition.

The idea behind Rayo’s number is something like “The biggest number you can single out of all finite numbers with an explanation of at most 10^(100) letters/numbers.” Possible such explanations could be “1000 to the power of 1000”, or “number of possible orders for a deck of 1000 different cards”. Since you have only finitely many symbols to use, there are only finitely many numbers you can single out like this, so there has to be a biggest one. Right?

Except the above has a problem. Because one valid singling out would be “One plus the biggest number you can single out of all finite numbers with an explanation of at most 10^(100) letters/numbers.” And this leads to a paradox. Hence, to make this idea work, you need to ask the answer to be in a language that doesn’t allow these kinds of paradoxes. One option would be to ask for answers in a programming language: ideas like this lead to the “busy beaver” function, which is one known way to produce huge numbers.

Rayo’s approach instead uses set theory and mathematical logic as the language. It stops at so called “first order logic”, which does not include statements like “for every possible formula” which could lead to issues like the above. So the slightly more accurate explanation is something like “The biggest number you can single out with a basic matematical logic formula of at most 10^(100) symbols.” (Or technically that plus one, as the actual definition is in terms of “smallest one you cannot specify anything bigger than” instead.)

It’s a very large number with no particular significance aside from being very large.

Roughly speaking, it says:

* Given a set of logical tools to make statements about numbers and constants, you can uniquely define a lot of numbers. Now, limit yourself to writing only definitions of at most 10^100 symbols in that language. This is a finite number of statements, so they can only define finitely many numbers. Rayo’s number is one more than the largest number they can define.

This definition turns out to result in an absolutely enormous number.