Ah but the real story is that like sophisticated math systems, in relation to any fixed axiomization, set theory involves provably, provably false and unprovable/independent statements. And in relation to any fixed model, set theory involves true and provable, true but unprovable, false but un-disprovable and false and disprovable.

And when you're math as a human endeavor, you can also add "provable but not yet proven" and "proven independent"

So, learn some stuff, see how far from black and white higher math can be.