Curiously enough, the Downward Löwenheim-Skolem theorem proves that if any first-order theory has an infinite model, then it also has a countable model, assumin

Curiously enough, the Downward Löwenheim-Skolem theorem proves that if any first-order theory has an infinite model, then it also has a countable model, assuming the theory itself is countable. So even though ZFC talks about uncountable real numbers, there is actually a countable model of ZFC that satisfies the same axioms. This isn’t an inconsistency, but it really does blur the lines between what we think of as mathematical objects that are “out there” in some sense (like uncomputable real numbers) or whether mathematics is just a game of symbol manipulation that also happens to predict the physical world pretty well.