It is an amazing and well-known fact that the Continuum Hypothesis is logically independent of Zermelo-Frankel set theory with the Axiom of Choice (ZFC), assuming it is consistent. In a similar vein, the Axiom of Choice itself is logically independent of Zermelo-Frankel set theory (ZF), assuming it is consistent.
I'm not well-versed in axiomatic set theory. In my mind, the set theory you choose (ZF, ZFC or some other one) defines for you the rules of the game we call mathematics. To say that a statement is independent of the given set theory is to say that the statement defines a new rule which is not a combination of the previous rules (so it is genuinely a new rule), but also forbidding what new rule allows is not a combination of the old rules (so you can't be in the situation that the new set of rules contradicts themselves unless they contradicted themselves to begin with). This may be a very coarse (and possibly inaccurate) view of set theory, but I hope it is not too much of an oversimplification for the question I have in mind.
I was trying to think of a simple mathematical analogy for the notion of a logically independent statement and I came up with the following:
Let $x \in (-1, 1]$. The statement "$x \in (-1, 0]$" is logically independent because neither it, nor its negation ("$x \in (0, 1]$") can be logically deduced from the given information (i.e. the analogue of a set theory).
Is this even close to being representative of the notion of logically independent? If not, why not? If it is, what details does it overlook?
Links to the relevant terms (it looked very strange if I linked to them all in the paragraph):