The question at hand here is that X can not be proven impossible, therefore it must be possible. Given that we are talking about things that happen in Nature, or more generally, the universe, X can in general be arbitrarily complex, and you cannot resolve X and all its logical and practical consequences by reducing X to just a label. Thus, we must regard X as such, namely something that cannot be trivially explained, but requires a quite complex linguistic and logical apparatus to resolve, as well as deep insight into the physical properties of X. Enter Gödel’s first incompleteness theorem;
Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e. there are statements of the language of F which can neither be proved nor disproved in F.
It is well known that “ordinary” language and logic that we use is sufficiently complex for the theorem to be applied, as Gödel himself proved the nonprovability of a sentence such as “This sentence can not be proved”. Granted, the sentence is quite clearly indirectly self-referential, but that does not rule out that statements about X are or must be self-referential in some way. In any case, whether or not X directly or indirecly imply self-reference does not matter, as the theorem will still apply. Thus, the sentence “It is impossible that X can exist” might not be possible to prove at all. However, under the incompleteness theorem, that does not rule out its validity. Thus, it does not follow that even if you cannot prove “it is impossible that X can exist” to be true, it does not mean it isn’t true. Thus, you can therefore not logically conclude that it is possible that X exists. All you are left with is an indeterminate answer.
In any case, the whole binary impossible vs possible concept seems of vanishingly little value to me, and does not really tell us much about Nature or Cosmos. Or about anything, really. Even if we manage to calculate the probability for something to be possible/to happen/to be true to P(something extremely unlikely) < 1/K, where K is some ridiculously large number, such as Graham’s number g64 or TREE(3) or g64TREE(3), or Γ(g64TREE(3)), P would still not be zero, and thus “possible”. What will be of value is that instead of the strictly binary choice, is that we attach probabilities to it, as has been argued before. The range could e.g. go from P(proven impossible) = 0 to P(proven to exist/happen/whatever) = 1. Even if you just restrict yourself to a limited discrete choice of P values, such as P ∈ {0, 0.000001, 0.001, 0.1, 0.25, 0.5, 0.75, 0.9, 0.99, 0.99999, 1}, it would be immensely more useful.