Eli5- what exactly is Rayo’s number?
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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.)
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