The problem is that "the number line" is ambiguous. Different people mean different things by it. Even any one individual's concept of it is inherently fuzzy, conflating innumerable slightly different meanings. For example, does it only contain real numbers? Or is every real number orbited by an infinite retinue of planet-like infinitesimals, each of which is accompanied by its own second-order satellites, the squares of the infinitesimals, at infinitum? If there are infinitesimals there must also be reciprocals of them, which are numbers which can't be reached by counting. Since I'm a computer person, the opposite approach is more attractive to me: Only computable numbers exist. Unless a finite-sized program with some finite amount of memory can, before the end of time, calculate the number to any desired finite precision, the number has a rather ghostly existence. The real numbers that make Cantor's diagonal argument work can only be created by flipping a coin infinitely many times to decide what each of its binary digits will be. And the resulting number can't be stored anywhere. We can talk about such numbers as a class, but there's no way to point to any one of them. Hence my post here one April Fools' Day when I made the absurd claim that I would be maintaining an online database of all uncomputable numbers. Computable numbers have pretty much all properties you'd want from reals. They're dense, they form a field, etc. If you have a task that requires a number than can never be calculated, you'd better find a different task. That's not quite right. There are numbers which are well-defined but uncomputable, e.g. the Chaitin constants and ... well, actually, those are the only counterexample I can think of. And since we can never know their values, we can live without them. If there are no infinities that can't be put into one-to-one correspondence with the integers, the continuum hypothesis is moot. Computable numbers can be put into one-to-one correspondence with the integers since computer programs basically *are* integers. I don't know of any model of quantum mechanics, or of anything else physical, which depends on CH being true of false. As for whether math is invented or discovered, obviously the latter, otherwise different people wouldn't get the same answers, any more than different artists produce identical paintings.