Let’s say a conjecture suggests for all $x$ in an infinite domain, $P(x)$ is true(conjectures such as the Goldbach Conjecture, the Collatz Conjecture, etc). Now, could we say no such conjecture could be unprovable? Here’s my reasoning: if such conjecture was unprovable, then wouldn’t that mean we couldn’t find a counter example and thus, $P(x)$ being true for all $x$? The reason I say we could find a counter example is because we assume it’s unprovable and if we did find a counter example, it would be provable.
P.S: I think I don’t really know what it means for something to be unprovable and that's where my mistake lies. So could you please explain a bit if that is indeed the case?
The issue is the possible complexity of $P$: how hard is it to tell whether something is in fact a counterexample to a given conjecture?
For example, take Goldbach. A putative counterexample $n$ to GC can be verified quite easily by just checking all the sums of primes smaller than $n$. So if our "background theory" is sufficiently strong to do basic arithmetic, then if Goldbach is not disprovable in that theory then Goldbach is true.
However, let's look at the Twin Prime Conjecture. Here a counterexample would be some $n$ such that there are no twin primes $>n$. But in order to verify this, in principle we would need to check infinitely many values. So perhaps there is a counterexample, but our axiom system - while strong enough to do basic arithmetic - is not strong enough to actually perform the desired verification in this case.
The relevant notion of complexity here is given by the arithmetical hierarchy, and roughly speaking we have that the following are equivalent (fixing a background axiom system which is appropriate - say, (first-order) Peano arithmetic):
If $P$ is not disprovable, then $P$ is true.
$P\leftrightarrow Q$ is provable for some $\Pi^0_1$ sentence $Q$ (note that $P$ itself might not be $\Pi^0_1$ - it could appear very complicated at first glance).
Now that said, we still can have a true $\Pi^0_1$ sentence be undecidable in our theory - the reason being that our theory is not able to tell that it is undecidable! E.g. the Godel sentence for PA, $G_{PA}$, is a true $Pi^0_1$ sentence which is undecidable in PA, but PA can't prove "PA doesn't prove $G_{PA}$;" all PA can prove is "If PA is consistent then PA doesn't prove $G_{PA}$," which isn't quite enough.