Could the Riemann hypothesis be provably unprovable? I don't know much about foundations and logic, so I ask forgiveness if my question is just plain stupid.
Assume we have a statement of the form:

There exist no $x\in X$ such that $P(x)$.

where $X$ is some kind of set (or class, or similar stuff) and $P$ is a set of properties. An example of such a statement could be the Riemann hypothesis:

There exist no $x\in\mathbb{C}$ such that $\Re(x)\neq\frac{1}{2}$ such that $x$ is not a negative even integer and $\zeta(x)=0$.

Can such a statement be provably unprovable?
Intuitively, I would say no, because to show that it is unprovable we should show that we cannot find $x\in X$ such that $P(x)$ (else finding such an $x$ would be a proof that the statement is false), but doing so would prove the statement to be true.
Is this correct, or am I missing something?

Edit: Please read the question correctly: it is not properly a question on the RH, but more a question on logic!

Let us say that there exists some $x \in \Bbb{C}$ such that $x$ is neither a negative integer nor on the line of $\Re(x) = \frac{1}{2}$.  Let us further suppose that some inspired mathematician conjectured that $x$ could be expressed as the sum of some convergent but some god-awful complicated series.
Now "experiemental mathematicians" might well look at this $x$ and determine that 
$\zeta(x)$ is zero to a thousand decimal places; but of course that does not constitute a proof that $\zeta(x)=0$.  And it is certainly logically plausible that the theorem 
"$\zeta(x)$, with $x$ defined as $\sum_\infty (\mbox{this horribly complicated mess})$, is zero" could be true but unprovable. 
So it is plausible that the Riemann hypothesis could be undecidable within the usual mathematical axiomatic framework.  Contrast this with Fermat's last theorem (ignoring Wile's proof): there, any given purported counterexample can be tested algorithmically in a finite number of steps, so if you could prove it was undecidable, you would be proving it false.  (This argument would not say it can't be undecidable, just that it cannot be proven to be undecidable.)
