So, the problem with Rayo's number is not what first order set theory to use; only the language is used, which is the same in all first order set theories.

The issue is Rayo does not specify what model of second order set theory he is using.

Whether he "should have" or not is an interesting philosophical question. For example, platonists would say that you could use a model where the elements are sets and $\in$ is $\in$, since they believe sets exist a priori.

A formalist, on the other hand, would not accept such a model a priori.

So, I guess in summary, rayo's definition takes a model of second order set theory and spits out a number from 1 to infinity. Rayo didn't specify a model though, so whether you or not think Rayo "implied" a model or not solves the question.

Really, the Big Number Duel was not terribly rigorous, since they never specified what background mathematical theory would be used to compare the numbers, since they quickly exceeded the strength of PA, for example.

Not even TA would work, since although they were talking about numbers, the definitions where not in the language of arithmetic.

So yeah.

Oh, and you might be wondering how Rayo's number is able to use only the language of first order set theory. If you look in Sat, however, you'll see that he is just analyzing its raw syntax.

we start with the weakest theory which is believed to be "absolutely true", then use it to derive the more "useful" theory and so on, thus extending the hierarchy indefinitely.

My understanding of Rayo's number was that the entire hierarchy of theories is required to be included in the formula somehow. But I still don't know the details of such a process...

Rayo's number is just a formula in second order set theory with one free variable, essentially. You can not even talk about it until you "hit" a theory in the language of second order set theory in the hierarchy.

The main question is if you belief that all statements in second order set theory are absolutely true or false. If you do, Rayo's number makes sense. Otherwise, it does not.

It may seem weird to think that it has an absolute meaning, but think of "2+2=4". Is the universally true or false? If so, what's the difference between that and Rayo's number.

I.e. when stating "2+2=4", should we require mathematicians to state what theory they are working in?

"think of "2+2=4". Is the universally true or false?" < It depends on the meaning of symbols "2", "4", "+" and "=", but as far as I understand, the fact that 2+2=4 is provable (in some chosen consistent set theory), but Rayo's number is believed to be so large because it takes into account unprovable statements.

You can prove that Rayo's number exists in kelly-morse set theory.

Of course, its value will depend on what model of KM you choose, but all models prove it exists.

Like, here is a simplified version of Rayo's number.

We say that model of first order set theory over a theory T is sound if it statements in it are true iff they are true using tarski's definition.

Rayo's number is the greatest number n such that there a string of at most googol symbols, called phi, such that phi defines x in any sound model of first order set theory for any theory T.

Thank you for the explanation! I'll try to explain my concern.

On the one hand, I see the following statement: "Rayo's number is provably a specific single number, defined by Rayo's definition" (source at Math.SE).

On the other hand, I see the following statement: "whether these definitions work or not seems to hinge on some very delicate issues about definability in set theory" (source at Mathoverflow).

The problem is that these two statements are incompatible. The number is either based on someone's personal beliefs or provably a specific single number (even it is impossible to determine the exact value).

The quote means that in most theories of second order set theory, you can prove Rayo's number is well defined.

Actually, some, not most.

The second quote is probably talking about NBG, another second order set theory in which Rayo's definition fails, I'm pretty sure.

Actually, wait, what does it do in NBG?

Oh, I think its independent of NBG, actually. It can't fail, since NBG is a sub theory of KM.

Actually wait no, it still works in NBG, it just can not prove many things about it.

Oh, and Scott Aaronson (the author of the second link) seems to require that a definition not depend on controversial statements about set theory for it to "work".

Rayo's number definitely could be affected by things like the continuum hypothesis.

Anyways, my personal philosophy is that there is an intended model of first order arithmetic, but not of set theory (first or second order), so Rayo's number would not be well defined if you do not specify a model. However, KM disagrees with me, and happily asserts that Rayo's number is a specific number, despite not knowing what it is.

Well, there is a subtly there. Unlike me, KM can't talk about the semantics of second order set theory.

"Rayo's number definitely could be affected by things like the continuum hypothesis." < Wait, but the continuum hypothesis is part of what? If it's included in the length of the formula, then I am not sure that it affects the value of Rayo's number. If it's not included in the length of the formula, then it means that the value of Rayo's number depends on the mathematical background, which totally breaks the elegance of the definition (and contradicts the rules of the game).

It is included in the length, but it could affect what value a given definition defines. For example the statement "x = 0 if CH and x = 1 if not CH" defines a number, but the number it defines depends on CH.

"it could affect what value a given definition defines." < Yes, but only for "a given definition" (a given formula)! Note that Rayo's number takes into account all possible formulas (that don't exceed some length) and takes the maximum value.

Well yeah, but maybe the maximum's value is decreased for some reason if you "toggle" CH, which would result in a new max potentially.

In particular, I think you could construct a nonstandard model in which Rayo's number is quite low.