Question about the Continuum Hypothesis The Continuum Hypothesis hypothesises

There is no set whose cardinality is strictly between that of the integers and the real numbers. 

Clearly this is either true or false - there either exists such a set, or there does not exist such a set. 
Paul Cohen proved that the Continuum Hypothesis cannot be proven or disproven using the axioms of ZFC.
If we find a set whose cardinality lies strictly between that of $\mathbb{N}$ and $\mathbb{R}$, then we are done, we have disproven it. But it has been proven that we cannot disprove it, thus by contrapositive, we cannot find such a set. If we cannot find such a set, then we can only conclude that such a set does not exist (if it did exist, there must be a non-zero probability that we would find it, so given enough time we would - contradiction. □)
So I have proven that the Continuum Hypothesis is true - there does not exist such a set. But this is a contradiction because it has been proven that we cannot prove it either. So where did I go wrong?
Thanks!
 A: "But it has been proven that we cannot disprove it" is not true. What has been proven is that you can't prove or disprove it within ZFC, or as you put it, "using the axioms of ZFC." You could easily prove it -- with a one-line proof, in fact! -- by adjoining to ZFC a new axiom, namely the Continuum Hypothesis.
(In other words, you need to pay closer attention to what it means to prove something within a specific formal system. In metalogic, theorems are relative to what axioms you start with. It makes no sense to speak of proving or disproving something, tout court.)
You'd also be rather hard-pressed to explain to a probabilist what probability measure allows you to say "if it did exist, there must be a non-zero probability that we would find it."
EDIT: you may find this paper by Solomon Feferman, entitled "Does mathematics need new axioms?", a fairly gentle introduction to some of the larger issues at stake here, which Andres has sketched in his answer.
A: 
Clearly this is either true or false - there either exists such a set, or there does not exist such a set.

This is the crux of the issue here. Gödel's completeness theorem tells us that a theory T proves a sentence $\varphi(\overline{x})$ if and only if $\varphi(\overline{x})$ holds in every model of T. The problem here is that if ZFC is consistent, then it has an enormous number of models, while you've implicitly asserted that there is some specific model, some structure, which is ZFC. This is simply not the case. How Cohen's proof works is that he shows that given some model of ZFC, you can both construct a model that has a new set that violates CH, or you can create a model that no longer believes it has such a set. 
A: A nice analogy is this: from the axioms which define a ring, it is impossible to prove that multiplication is commutative. "Preposterous!", you  might claim. "Take your ring to be the integers! Now, multiplication is repeated addition, and addition is commutative because we are working in a ring, and so perhaps by induction or maybe something simpler, multiplication is commutative!" 
But if instead we use matrices as our ring, I can find you some rather obvious counterexamples. It turns out that the axioms which define a ring are not strong enough to prove or disprove the commutativity of multiplication—we can have rings where multiplication is commutative, and other rings where it is not. Similarly, we can construct worlds according to the axioms of ZFC in which the continuum hypothesis holds, and others in which it does not, hence proving that the ZFC axioms are not strong enough to prove the continuum hypothesis. 
A: Another problem is that because just because a set exists doesn't mean you can find it.
A: You have a mistake here: "if we cannot find such a set, then we can only conclude that such a set does not exist".
That's not right. The fact that you can't find a set does not mean that it does not exist. That's like the planets located beyond the observable universe: you can't prove if there are some people like us over there or not (or whatever), even showing a particular case, because you can't ever look at these planets, and this does not mean they do not exist.
The Gödel-Cohen demonstration states that starting from these axioms you will never be able to assert that 'there is at least a set -that I don't know which one it is- that has a strict intermediary cardinality', and that you'll never be able to assert that such a set cannot exist. That is: no 'deductive' prove can ever state that it exists or that it doesn't. So, there is no need to try this kind of demonstration. But the question about this hypothesis remains always open (that's why it is a hypothesis): there could exist such a set, or there could not. If it exists and you can write it, you just can't prove its cardinality with the only help of the Zermelo-Fraenkel axioms. It could exist but no one could prove it. Why not?
This would not necessarily apply starting from a different (and not equivalent) set of axioms. But the 'logic world' that create these axioms (or any other axiomatic system that includes them) is always uncomplete: not every true statement can be proven. The "provable statements" is a subset of the "true statements". There is also an "observable universe" limit in maths. That's what Gödel demonstrated.
A: Cohen's result is that from a certain set of axioms ($\mathsf{ZFC}$) we cannot prove the continuum hypothesis. Gödel's result is that from the same set of axioms, we cannot refute the continuum hypothesis. This only means that the set of axioms under consideration is not strong enough to settle this question. If you manage to exhibit a set of intermediate size, your argument necessarily uses axioms beyond those in $\mathsf{ZFC}$, or is not formalizable in first-order logic. Similarly if you manage to show that there is no such set. It is worth pointing out that there are standard axioms beyond those in $\mathsf{ZFC}$ that settle the continuum problem. These axioms are not universally accepted yet, but this illustrates that, as explained, their results are not absolute, but relative to a very specific background theory.
