If in some theory it is impossible to prove $X$, is it impossible to disprove $X$ too?

Question

In mathematical logic, if we can prove that it is impossible to prove a statement $X$ with some theory $T$, do we necessarily have a proof that it is impossible to disprove the statement $X$ in theory $T$?
Now, this is obviously not right, but my question is rather about how should I understand statements/results that say it is impossible to prove something:

I was reading the Wikipedia page on 'Measurable cardinals' and it is stated there that:

the existence of a measurable cardinal is not provable in ZFC.

Does it then also mean that I can't find a proof that a measurable cardinal cannot exist? Because, if I could find a proof that it doesn't exist then obviously I can't find a proof that it exists (or else ZFC is inconsistent). That is consistent with the statement from Wikipedia:

"There isn't a proof that it exists".

So the statement from Wikipedia doesn't say that there isn't a proof that there isn't a measurable cardinal. But some common sense and natural language processing tells me that the sentence 'existence of measurable cardinal is...' accounts both statements 'it exists' and 'it doesn't exist'. I am deeply confused.

Do theorems that say it is impossible to prove something naturally mean the same as the statement is undecidable? Because they shouldn't, but they sound like they do:
If it said "It is impossible to prove it exists", it would be clear;
If it said "It is impossible to answer whether it exists or not", it would be clear again, but
when it says: "The existence is impossible to be proven" isn't so clear to me. It could mean both of previous statements and they are not equivalent.

Answer 1

Phrase "existence of a measurable cardinal is not provable" means "there is no proof that (a measurable cardinal exists)". It claims nothing about existence of proof that (a measurable cardinal doesn't exist). For example, it is also correct to say "there is no proof that (set of all sets exists)".

But as in general we are working in a consistence theory, it's trivial that if we can disprove $X$, then we can't prove $X$. So if $X$ is disproved, we don't usually say "we can't prove $X$" - it's true, but obvious.

Sometimes we know that $X$ can't be proved, but don't know yet if it can be disproved, but I think results like this are even more rare than "$X$ can be neither proved nor disproved".

Answer 2

Many example of unprovable statements are of the form "The computation of this Turing machine will never stop". One way to see that this includes interesting examples is given by the Turing machine that searches for a proof of $0 = 1$ in $\mathrm{ZFC}$.

The negation of "The computation of this Turing machine will never stop" is "There is a time $t \in \mathbb{N}$ at which the computation has halted". If this is true, we can always prove it by showing the computation up to time $t$. Thus, statements of the form "The computation of this Turing machine will never stop" can be

1. true and we can prove it
2. true but unprovable
3. false and we can prove it

but not actually false, but we cannot prove that they are false.