Veritasium has a nice video on Godel’s incompleteness theorem. It’s quite ironic that it was mathematically provable that analytic systems can never be complete, and can’t be provably coherent.
https://m.youtube.com/watch?v=HeQX2HjkcNo&pp=ygUUdmVyaXRhc2l...
If you somehow find an unprovable theorem (quite rare), you can always try with a different set of axioms. Mathematics are not about proving absolute truths of the universe but rather of pushing reasoning over a set of axioms the furthest possible.
Also as a nitpick, analytic systems that don't contain arithmetic can be complete and proven coherent.
As for the real world, facts that can neither be proven true nor false (existence of an immaterial soul for example), I think should be left at that. It is useless speculating about things we can't ever know of. I leave that to religion/spirituality.
And how do you know which things we can know of and which we cannot? Trusting your gut instinct on that isn't scientific. And why is speculating about immaterial things useless? I'm sure many great mathematicians heard some form of "what you're doing is useless and has no use or relation in the real world" especially in the realm of pure mathematics.
You bring up the soul which is a convenient example, but let's instead use a concept which YOU know exists for yourself which is consciousness. Can we ever know anything more about the mystery of consciousness or life or why any of this world and universe exists? Are those unknowable? Should we not talk about them? Should we only try to apply the lens of science here and for some reason not try to advance our understanding using philosophy even though it might not be as formal and unambiguous as math?
Imagine reality as the problem space of all things that could exist within the constraints of physics. The problem with observational evidence that it is only providing a tiny window into what is possible, really only the most probable are going to be what you see for the most part. Philosophy gives a means of meta views of systems and simplified system views that allow us to find otherwise unreachable islands of what can exist in our reality.
As for the other questions you mentioned, I still haven't found any reason to believe their answers are unkowable.
If anyone disproves my first reasoning, I will have to consider the question of the soul as worth pursuing again.
Just like in mathematics, if you have sufficient proof that a theorem is unprovable, it is useless trying to prove it!
So no, I don't trust my gut feeling about wether or not to seek the answer to something.
Not death of logic, since logic is still consistent. Gödel didn't say math was wrong, just incomplete.
So philosophy might step into the space beyond where a current logical set of axioms can reach. To explore something un-provable and maybe find a new set of axioms that were not reachable through logic built on the previous axioms.