I do not pretend to know much about mathematical logic. But my curiosity was piqued when I read Hofstadter's Gödel, Escher, Bach, which tries to explain the proof of Gödel's first incompleteness theorem by using an invented formal system called TNT--Typographical Number Theory.

Hofstadter shows how we are able to construct an undecidable TNT string, which is a statement about number theory on one level, and asserts its own underivability within TNT on another level.

My question is: Has this statement about number theory, call it the Gödel sentence, ever been explicitly found? Of course, it would be enormously long. But it seems possible to write a computer program to print it out. I know what is important is that such a string exists, but it would be fun to see the actual string, and it seems like a nice coding problem.

Forgive me if this is a naïve question.

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    $\begingroup$ How much paper/computer memory do you have? $\endgroup$ – Henry Jul 8 '14 at 23:25
  • $\begingroup$ I once saw a web page that purported to have the Gödel number on it, but did not have the details of the encoding and the axioms used. In base $10$ it about filled a screen. $\endgroup$ – Ross Millikan Jul 8 '14 at 23:52
  • $\begingroup$ The question supposes there is just one Gödel sentence. But producing the sentence requires choosing one particular Gödel numbering system, as well as various arbitrary choices regarding the syntax of the object language, and the proof theory. There are various different Gödel sentences depending on these choices. This makes the construction of just one of them less interesting, I would think. $\endgroup$ – MikeC Jul 9 '14 at 16:38
  • $\begingroup$ I think most proofs that the Gödel sentence exists are explicit constructions -- it's just that nobody bothers to ever actually do the arithmetic. $\endgroup$ – Hurkyl Aug 30 '17 at 12:32

I have once played around with this stuff myself and obtained this example of such a sentence. The long thing at the end of that page (there are in fact two long things, just minor variants if I recall it correctly).

  • $\begingroup$ Interesting! That is exactly the sort of thing I was seeking. $\endgroup$ – StrangerLoop Jul 8 '14 at 23:48
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    $\begingroup$ There is of ocurse no full warranty of correctness. But interestingly, the latin alphabet was just long enough to supply enough variables ;) $\endgroup$ – Hagen von Eitzen Jul 8 '14 at 23:53
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    $\begingroup$ can you translate your page into english? $\endgroup$ – Willemien Jul 9 '14 at 8:46
  • $\begingroup$ @HagenvonEitzen What are the axioms of TNT which the formula uses? I suspect these are ones of Peano arithmetic, but I wasn't able to track down where there is something said about induction axioms. $\endgroup$ – Wojowu Oct 30 '15 at 18:41

I was searching for an explicit representation of the Gödel sentence as well, and I found one today, only after reading your question here, so I share what I found: https://web.archive.org/web/20160528092209/http://tachyos.org/godel/Godel_number.html

Unfortunately it is not mentioned, what is the precise theory, what are the coding numbers of symbols, and what are the coding/decoding functions (hopefully these are consistent with Chaitin's book mentioned on right of the page), so the number is not really meaningful in its own, but at least gives an order of magnitude (if it is correct).


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