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I want to start a project to explicitly generate the Gödel sentence on the incompleteness theorem (Rosser version, Robison Arithmetic, with Gödel numbers in it -- the full thing). I read some proofs of it, Smullyan on its Incompleteness Theorem book, and Boolos Jeffrey Burgess proof. I think the problem to generate it with the help of a average modern computer is the beta function. Beta functions based on the Chinese theorem seems to be an impossible choice. Now, I think Smullyan beta function seems feasible (p.45, the way I see it, it seems to be based on a kind of ASCII-string code). So here are the questions:

  1. What is the better approach to construct this beta function? (do you have any reference? etc...)
  2. Is the Smullyan way feasible?
  3. Do you have some tips?
  4. Do you think Python is good for it?

Ok, I know this has a lot of details, and it is required some programming skills. I know some Python, C, I am aiming to learn more, so this is a long term project. I was looking more tips just to not take the wrong direction.

Ps: I noticed there is this aswer, but I was more looking for some advice rather than the sentence.

Lost definition
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    "I think the problem to generate it with the help of a average modern computer is the beta function. Beta functions based on the Chinese theorem seems to be an impossible choice." Why do you say this? I don't see any problem implementing the standard $\beta$ function. – Noah Schweber May 21 '21 at 17:54
  • Why do you want to do this? There is nothing canonical or interesting about a number that represents a particular formula. – Rob Arthan May 21 '21 at 20:37
  • @NoahSchweber, hi, I will use this setup of proof to show what I am thinking. Just to take a example. Take $a_0 = 4, a_1 =6$. Then the least $m$ that will do the work is $m = 4$, that implies $m_0 = 5$ and $m_1 = 9$. The first solution to the Chinese system is $x_0 = 24$. Then $s = \pi (24, 4) = 410$ (standard pairing). Well $s=410$ is too big. And with the magnitude of the Gödel numbers it will be enormous, and I don't know if it can solve the Chinese System with this numbers. – Lost definition May 23 '21 at 01:12
  • I mean, it will have to do this operation multiple times, for what I am remembering one inside the other, and I don't even know where it will reach. I really don't know. But if it is feasible, it would be very nice, because I think it is the most elegant beta function. – Lost definition May 23 '21 at 01:17
  • @RobArthan yes, it seems there is nothing interesting about this number, of course it depends on the Gödel numbering and the language of the formal sytsem. But it is not that what I want. I want to do it for didactic reasons. I think it will help me in two ways: understand better the incompleteness theorem, and improve my programming. Now, I don't know if it is a nice thing. I mean, I would love to read a book that gives me details about pratical programming and logic at the same time. And I think some books forgets that we have good computers now. Do you think it would be nice to do this? – Lost definition May 23 '21 at 01:34
  • @Lostdefinition: whatever turns you on! You may find in John Harrison's "Handbook of Practical Logic and Automated Reasoning" useful as it contains lots of examples of practical programming with logical applications. The section on the incompleteness theorem may be particularly relevant. – Rob Arthan May 23 '21 at 12:59
  • "Well $s=410$ is too big." Is it? What exactly are the relevant thresholds, here? I don't see that it is too big, offhand. – Noah Schweber May 23 '21 at 17:24
  • @NoahSchweber, hmmm, I will try to play with it and see what are the thresholds, because I have no idea. In fact, now I think I misunderstood some things about it. I will try to revisit it with calm. – Lost definition May 24 '21 at 16:49
  • @RobArthan nice, I will look into it. – Lost definition May 24 '21 at 17:02

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