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Rayo number is named by an expression in the language of first order set-theory. I wonder if we extend this definition to any higher order logics, will the result significantly change? (because for most (useful) models in higher logic we can still find expression in first order logic)

I am not professional in set theory/proof theory. But I will try to understand professional answers. Thank you!

APPLSA
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Yes. In general, given any logic $\mathcal{L}$ with a notion of "length of formula" according to which every formula is finite and only finitely many formulas have a given length (not all logics have such length notions!) and any "ambient structure" $\mathfrak{A}$ containing the naturals, the Rayo function for $\mathcal{L}$ in $\mathfrak{A}$ makes sense. This is the function sending $n$ to the smallest number greater than every $k$ such that $k$ is definable in $\mathfrak{A}$ by an $\mathcal{L}$-formula of length $\le n$. See here for a summary of definability in the context of first-order logic; the basics remain unchanged when we replace first-order logic with some other logic, and an element is definable iff (by definition) the singleton containing it is definable.

Rayo's function as usually defined takes $\mathfrak{A}$ to be the whole set-theoretic universe $V$. This isn't properly speaking a structure, since it's too big, and this is what saves us from the apparent paradox. Similarly, in order to make sense of the Rayo function for $\mathcal{L}$ in $V$ we'll need to cheat and work in some "even larger" mathematical system. But ignoring this issue for the moment, the answer to your question is definitely yes: changing the logic can dramatically change the corresponding function.

Here's one easy observation: since in any "reasonable" structure $\mathfrak{A}$, first-order truth is second-order definable, the Rayo function for $\mathsf{SOL}$ (= second-order logic) in $\mathfrak{A}$ will eventually dominate the Rayo function for first-order logic in $\mathfrak{A}$. In fact more is true: the latter won't even be $O$-of-the-former.

Noah Schweber
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  • Presumably you can also do something like that in $\cal L_{\kappa,\omega}$ wherein you can define the rank of a formula, no? – Asaf Karagila Oct 21 '22 at 09:24
  • @AsafKaragila Well at that point you need to talk about ordinals rather than just natural numbers in terms of the codomain of the appropriate Rayo analogue, but yes. – Noah Schweber Oct 21 '22 at 17:13
  • Clearly. It does raise an interesting question about $\cal L_{\infty,\omega}$, and the Rayo number it defines. – Asaf Karagila Oct 21 '22 at 17:51
  • @AsafKaragila Indeed! – Noah Schweber Oct 21 '22 at 17:58
  • According to this comment at The Blog of Scott Aaronson, Agustin Rayo (who invented Rayo’s number) "was clear and explicit that, if you reject the idea of an “intended model of ZFC” (something that would even decide, e.g., the Axiom of Choice and Continuum Hypothesis), then you should also reject the well-definedness of his number." – lyrically wicked Oct 22 '22 at 09:24
  • According to this article at Googology Wiki, in 2020, Rayo added a new description of the way to deal with Rayo's number, which "means that Rayo considers a philosophical "interpretation" of set theoretic formulae with respect to the "truth" in the real world, which is unformalisable in mathematics, and does not intend a specific choice of axioms." – lyrically wicked Oct 22 '22 at 09:34
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    "See here for a summary" — missing link? – lyrically wicked Oct 22 '22 at 09:44
  • @lyricallywicked Re: your last comment: whoops, fixed! Re: your previous two comments: I'm not sure what you're asking/saying. – Noah Schweber Oct 25 '22 at 16:42
  • @NoahSchweber: “Re: your previous two comments: I'm not sure what you're asking/saying” — I was trying to say that I have not yet seen a full technical description of how “the Rayo function $f$ for $\mathcal{L}$ in $V$” may look like. Note that the initial purpose of this function is to win the game. Then suppose that the opponents are professional set theorists who are determined to prove that a given description of $f$ is flawed. I would suppose that a lot of issues must be addressed if one wants to provide a satisfactory description of $f$. – lyrically wicked Oct 28 '22 at 04:31
  • Could you enlighten me a bit more about how to reason from "first-order truth is second-order definable" to the "Rayo function for SOL (= second-order logic) in A will eventually dominate the Rayo function for first-order logic in A"? – APPLSA Nov 19 '22 at 03:08
  • @NoahSchweber I should add a mention – APPLSA Nov 19 '22 at 09:12