When we try to solve the hard problem like 'ABC conjecture',should we first examine if it is a proposition that neither prove nor disprove

Question

In mathematic logic,Godel's incompleteness theorem tell us that a proposition that neither prove nor disprove is exist.so did every famous hard problem need to be examine first?

Answer 1

Let us take $\operatorname{ZFC}$ as our background theory and let us assume its consistency (for convenience).

You're right that Gödel's Incompleteness Theorem guarantees that there are statements $\phi$ whose truth value is not decided by our theory. In fact, over the last 50+ years (more than that, if we also consider the work by Mostowski and co on $\operatorname{ZFA}$) not only have we proved the relative independence of all kinds of statements, we have considerably improved our tools to do so. The most important tool is Cohen's method of forcing, originally introduced to prove the relative independence of the axiom of choice and the continuum hypothesis.

Another important tool to settle questions of independence is inner model theory. Sometimes we can prove (the consistency) of a certain statement $\phi$ by adding additional axioms to our theory. The question then becomes whether these additional axioms were necessary. Maybe there was a prove in $\operatorname{ZFC}$, but we weren't clever enough to find it? Inner model theory, sometimes, enables us to prove the necessity of certain additional axioms that provably don't follow from $\operatorname{ZFC}$. This is another method to prove the relative independence of $\phi$.

That being said, when it comes to 'classical' questions in mathematics, there aren't any convincing independence results that I know of. And there is a good reason for that. A consequence of Shoenfield's Absolutness Theorem is that forcing and certain methods of inner model theory cannot prove the relative independence of most classical questions, since these are usually of complexity at most $\Pi^1_2$ or $\Sigma^1_2$. This is true for $\operatorname{P} \text{ vs. } \operatorname{NP}$, the Riemann hypothesis, the ABC conjecture, the Goldbach conjecture, the Collatz conjecture, ...

The consensus seems to be that these questions are in fact not independent, but I'm not really interested in this sort of speculation. There are however two mathematical facts about these sorts of problem I am definitely interested in:

1. If one of these problems is independent, it seems like we need much finer control over our universe to prove this fact. Therefore, much like Cohen's work on forcing, settling the independence of one of these results would likely lead to a whole bunch of new and exciting mathematics. There is some ongoing work in this direction, but I know way too little to talk about it.
2. Another consequence of Shoenfield's absoluteness is that if we are able to answer one of these question via forcing or in $L$ or in an inner model that doesn't rely on so-called 'small large cardinals', then this already implies that it was provable in $\operatorname{ZFC}$ to begin with. This allows mathematicians to use a whole lot of additional machinery to tackle those questions without abandoning their philosophical (dis-)beliefs on certain set theoretical questions. Unfortunately, it seems, that this additional machinery isn't very applicable in these scenarios, but who knows? (I'd love a proof of, say, Riemann's Hypothesis to rely on $\Diamond$ in a fundamental way, i.e. in a way that it isn't clear how to get rid of this assumption without referring to some absoluteness argument).