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!
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.