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In Who can name the bigger number?, Scott Aaronson gives two examples of fast-growing functions:

  • The Ackermann sequence, defined specifically as A(1)=1+1, A(2)=2*2, A(3)=3^3, etc
  • The family of Busy Beaver functions.

Wikipedia also has examples of fast-growing functions, like the TREE sequence and the SCG function. All of these are discrete functions. Even "simpler" fast-growing functions, like $2 \uparrow^n 2$, are discrete. We could define continuous functions via interpolation but that feels unnatural.

Are there any very fast growing functions, faster at least than BB, that are also "naturally" continuous?

Hovercouch
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    One important thing to keep in mind is the difference in logical complexity between $\mathbb{N}$ and $\mathbb{R}$. As a structure the reals are much simpler than the naturals; to be precise, $\mathfrak{R}=(\mathbb{R};+,\times)$ is decidable but $\mathfrak{N}=(\mathbb{N};+,\times)$ isn't (note that the Busy Beaver function is definable over $\mathfrak{N}$). Moreover, $\mathfrak{R}$ stays decidable even when expanded with various functions of interest, e.g. $e^x$ or $\sin(x)$. I think this provides some (vague) evidence for the nonexistence of a function of the type you're looking for. – Noah Schweber Feb 16 '23 at 21:35
  • @NoahSchweber That has to mean that $\Bbb N$ can't be defined in $\Bbb R$ using a first-order formula (else the theory of $\Bbb N$ could be decided by checking in $\Bbb R$ whether any formula has a solution in $\Bbb N$). And since the positives can be defined in $\Bbb R$, that means $\Bbb Z$ can't be defined in $\Bbb R$. – Robert Shore Feb 17 '23 at 01:10
  • @RobertShore That is correct. FWIW $\mathbb{N}$ is (FO-)definable in $\mathbb{Z}$ (via Legendre's four square theorem) and even in $\mathbb{Q}$ (via a more difficult argument), see here. Real closed-ness is a surprisingly strong "taming" condition. (Incidentally, see the discussion here for some further subtleties about properly pinning down the theory of real closed fields.) – Noah Schweber Feb 17 '23 at 01:13
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    Re "fast-growing* functions, like $2\uparrow^n 2$"*: It happens that $2\uparrow^n 2=4$ for all $n\in\mathbb{N}.$ – r.e.s. Mar 05 '23 at 13:53
  • @NoahSchweber: Thanks for sharing about the decidability of R. That's new to me and very interesting. It looks like the extension to $(\mathbb{R}, +, \times, \pi, \ln(2), e^x, \sin)$ is undecidable according to Richardson's theorem https://en.wikipedia.org/wiki/Richardson%27s_theorem . Do you agree or am I misunderstanding that? (Not sure if that would help to define a Busy Beaver function over R ...) – sligocki May 07 '23 at 20:31

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