"But how would we then prove very unrelated concepts such as "the moon is made of cheese" or something similar to this? Or is "anything" really only within the context of the system we're working in?"
Or you can attempt the former if you're willing to do a lot of work, for example find the density of cheese and density of moon. Then Prove that the two numbers are equal. And then do the same with the moon's spectral analysis and that of the cheese, basically all physical characteristics. Then you can say that with overwhelming evidence, the moon is made of cheese.
Note that the proof that the moon is made of cheese (or that the moon is made of rocks for that matter) is no longer in the realm of "pure mathematics". That's physics and that requires empirical evidence... so we can only say with a high degree of certainty that the moon is made of rocks but we can never be 100% sure.