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.
