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I really have a hard time asking this question. Because my mathematical background is almost at school level. I do not know in which theories of mathematics these questions are addressed. Unfortunately, since my English is insufficient, I will use the simplest words to describe my question.

My question consists of $4$ parts:

Suppose that pure mathematical conjecture $X$ is given. It is never possible to prove that this conjecture is definitely true/correct. It is never possible to prove that this conjecture is definitely wrong/false. Can we deduce that this conjecture is definitely undecidable?

Suppose that pure mathematical conjecture $Y_1$ is given. It is never possible to prove that this conjecture is definitely true/correct. It is never possible to prove that this conjecture is definitely undecidable. Can we deduce that this conjecture is definitely wrong/false? Or, at least can we deduce that the Conjecture can be proved to be falsifiable?

Suppose that pure mathematical conjecture $Y_2$ is given. It is never possible to prove that this conjecture is definitely wrong/false. It is never possible to prove that this conjecture is definitely undecidable. Can we deduce that this conjecture is definitely true/correct? Or, at least can we deduce that the Conjecture can be proved to be verifiable?

Suppose that pure mathematical conjecture $Z$ is given. It is never possible to prove that this conjecture is definitely true/correct. It is never possible to prove that this conjecture is definitely wrong/false. It is never possible to prove that this conjecture is definitely undecidable. Is such a kind of conjecture possible? What is the logical status of this conjecture?

Finally, I mean with "pure mathematical conjecture ", for example $X/Y/Z$ can be Goldbach Conjecture/ Collatz Conjecture and etc.

I couldn't express my question as I wanted. (due to lack of grammar) But, I tried to choose the right words as much as I can.

Thank you very much!

Community
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lone student
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  • If there is a deficiency in the question, I invite you to comment. – lone student Mar 26 '20 at 01:05
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    The Question seems more philosophical than mathematical. How would you define undecidable? You seem to treat it as an absolute notion. You beg issues of knowledge by asking that "pure mathematical conjecture" such and such "is never possible to" ... but have not defined these notions in a way that allows your Question to be resolved with mathematical reasoning. In math definitions are your friends! – hardmath Mar 26 '20 at 03:32
  • @hardmath My question(s) is probably a few steps away from being meaningful. I don't know the logical definitions. But I know exactly what I want to ask. But I'm having trouble telling people. The problem is here. In this way it was rightly closed.. – lone student Mar 26 '20 at 15:33
  • Questions are often posted in unpolished form. Finding the right words can be a significant burden but also a significant contribution to learning. There is a formal notion of proof, one that is relative a formalized system of language, axioms, and rules of inference. So asking whether a statement is provable would (in the context of mathematical foundations) need to provide what the allowed formal system is. – hardmath Mar 26 '20 at 17:59
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    I think if you look at the way I worded things in my answer, lone, you can get some idea of how to phrase your question to meet the objections of those who voted to close. Pick an axiomatic system, such as Peano arithmetic or ZFC, one that is known to be strong enough to be incomplete (provided it is consistent), and rephrase your question in terms of proofs in that system. Alternatively, do a search on this site, as I suspect the questions you raise have been asked and answered before. https://math.stackexchange.com/questions/17212/is-there-a-statement-whose-undecidability-is-undecidable/ – Gerry Myerson Mar 27 '20 at 00:47
  • @GerryMyerson Thank you for your guidance. I will do what you say. But rather than editing my question, I will look for answers close to me. Thanks for the link. – lone student Mar 27 '20 at 07:35
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    A side remark. There is no conjecture $XYZ$, but by coincidence, a famous conjecture in number theory is the $abc$ conjecture. – J.-E. Pin Mar 28 '20 at 06:08
  • @J.-E.Pin Thank you for comment. It was interesting! – lone student Mar 29 '20 at 01:28

1 Answers1

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First of all, all proofs are proofs within some given axiomatic system. We can speak of a proposition being (or not being) provable in Peano Arithmetic, or in Zermelo-Frankel set theory, or first-order theory of groups, etc., and it's only in such a context that we can speak of provability.

Second, all discussions of decideability rest on the assumption that whichever axiomatic system we are working in is consistent, that is, the assumption that the system won't prove any contradictions. When people say such and such a statement is undecideable, that's shorthand for such and such a statement is undecideable in such and such a system, provided the system is consistent.

Now, if we can prove that, if our system is consistent, then there is no proof of X in our system, and no proof of the negation of X, then we have proved that (if our system is consistent, then) X is undecideable in that system.

Questions involving undecideability of undecideability make my head spin. I'll leave them to someone with better training than mine.

Gerry Myerson
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  • Thank you for Your answer. I am not sure that I expressed my questions well. But I did my best. – lone student Mar 26 '20 at 01:29
  • Can I ask one question? Does "undecideability of undecideability" imply the conjecture $Z$? – lone student Mar 26 '20 at 01:35
  • I don't think Z makes sense. As I wrote, if you prove that if your system is consistent then neither Z nor its negation is provable, then you have proved that if your system is consistent then Z in undecidable. So on the assumption of consistency, the situation you describe for Z is impossible. – Gerry Myerson Mar 26 '20 at 01:45
  • I understood.So, $Z$ is impossible. I am trying to understand your answer line by line. Finally, can you make a little detail about $ Y_1$ and $Y_2?$ if you have time? – lone student Mar 26 '20 at 02:02
  • Time isn't my problem. Lack of expertise, that's my problem. Don't worry, someone will come along who is better equipped than I am to puzzle thorugh $Y_1$ and $Y_2$. – Gerry Myerson Mar 26 '20 at 02:05
  • I know you are doing modesty.. By the way, I am not asking questions as an educated person. For this reason, I cannot prevent the questions from being seen as absurd. Thank you again for the answer. – lone student Mar 26 '20 at 02:15
  • @GerryMyerson: Concerning unprovable independence, you can have a look at my posts here (for some background on truth vs provability) and here (for the answer). It is really quite simple once we have the fixed-point theorem and abstract out the provability logic. – user21820 Aug 21 '21 at 09:50
  • @user, thanks, but it's still beyond me, or rather it's beyond the amount of effort I'm willing to expend to understand it. – Gerry Myerson Aug 21 '21 at 12:18