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I learned today that there are some true statements that are unprovable (gödel incompleteness theorem), which is a liddle sad. But not too sad if we could at least proof that:

For every unprovable true statement, exists a proof that it is unprovable.

Because that means we can stop looking.

So I wonder does something like that exists?

And if not, is there any other way to deal with it?

Xanlantos
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  • In fact, if $T$ is any essentially undecidable theory (basically, a theory to which Godel's argument applies), then the set of sentences which are $T$-independent is not computable since otherwise we could build a computable completion of $T$ via a kind of greedy algorithm. – Noah Schweber Jun 02 '21 at 14:45
  • Unfortunately the system won't let me add any more duplicates, but there are many. It is good to wish and contribute, but you should (1) search before posting, especially when you want to post a question and an answer at the same time; and perhaps more importantly, (2) it's a bit disingenuous to have deleted the question you asked on MathOverflow and post it here with parts of the answer you received there without any credit to the person who wrote it there. – Asaf Karagila Jun 02 '21 at 14:45
  • I just moved the question, nothing more nothing less. That was what I was told, that was what I did. I search for it... ofc I did, but as so often, if you are not proficient in a field asking the right questions is already hard to do. I tried to give him credit to, but I could not find a way to contact him after my question was deleted there. And so I decided to just copy and paste and do not worry about it any longer. At some point fitting in the SE community is a more monumental task than I feel its worth. I feel prejudiced way too often for totally valid and understandable mistakes. – Xanlantos Jun 02 '21 at 14:55
  • That I put it here, is solely to help others. I got my answer already. I was just trying to help. So If this is not considered helpful, I rather have it deleted. – Xanlantos Jun 02 '21 at 15:13

1 Answers1

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One thing to keep in mind is that "provable" is always with respect to some axiomatic system, such as Peano Arithmetic, or ZFC. And when we show something is unprovable, we have to do so in a larger system and reason about our initial system.

That said, we do know roughly speaking that there are statements which are unprovable and where we cannot prove that in any reasonable system.

Consider the Turing Halting problem. This asks whether given a specific Turing Machine, whether it will halt when run on a given input tape. We know that this problem is in general unsolvable, in the strong sense that there is no Turing machine which solves the Halting problem. However, note that if a given Turing machine halts on a given tape, then there is a proof of that: simply run the machine that many steps. Suppose now that there were some general axiomatic system A where we could always either prove that whether a given Turing machine halting on a given tape was undecidable in Peano Arithmetic (or whatever your preferred system of axioms is that is strong enough to talk about Turing machines).

Then, we solve the Halting Problem, since we could encode a Turing machine M, which given another Turing machine X and an input y, alternates between steps of running X in y, and searching for a proof in A, that X halting on y is undecidable in Peano Arithmetic. By assumption, this machine will eventually either find when X halts on y, or will find a proof in A that Peano Arithmetic will not resolve the problem, which means that X must never halt on y.

Thus, we have given the existence of A, constructed a Turing machine M which can solve the Halting problem. But we know that such a machine cannot exist, so our assumption that we had such an axiomatic system must be false.

(There are some subtleties here I've brushed under the rug. I've assumed consistency of all relevant systems, and we need to be careful about what we mean by an axiomatic system. But the central ideas here are correct.)

Thanks to JoshuaZ (feel free to post the answer yourself, I will accept it)

Xanlantos
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