statement that is consistent in ZFC but the negation of it can be both consistent and inconsistent in ZFC or vice versa Is there any known case where ZFC system is known to be consistent with a statement, but is also consistent with the negation of the statement? Or vice versa.
Also, when we say ZFC is consistent with something, are we saying that one particular model of ZFC allows something as being consistent?
Edit: Phrasing the question this way, I see I was somehow mixed up. Answers to two questions are obvious... For some checks, can anyone answer the second one? - though I think the answer is obviously.. (as some other models can behave differently)

Let's say you have a first-order theory $T$ and a sentence $\phi$. If $T$ is both consistent with $\phi$ and consistent with $\neg \phi$ then $\phi$ is independent of $T$, which means that if $T$ has a model, then it has models in which $\phi$ is satisfied and models in which $\neg \phi$ is satisfied.
Here we're considering $T=\text{ZFC}$. Let our $\phi$ be $\text{CH}$, which is the assertion that $2^{\aleph_0}=\aleph_1$ (this is known as the Continuum Hypothesis). Then there are methods which can be used to prove that
$$\text{ZFC is consistent}\ \Leftrightarrow\ \text{ZFC+CH is consistent}\ \Leftrightarrow\ \text{ZFC+(}\neg\text{CH) is consistent}$$
That is, if there is a model of ZFC, then there are models in which the continuum hypothesis holds and models in which its negation holds. (And, in fact, if there is a model in which CH holds then there is a model in which the negation of CH holds.)
