[ColinWright]

If you assume that there is no number bigger than zero but smaller than every positive number (basically the Archimedean property) then you can prove that "you've got them all." You use Dedekind cuts.

Suppose there's a location on the line that's somehow missing - call it x. Let A be all the numbers less than x, let B be all the numbers greater than x, and that gives you your Dedekind cut. That Dedekind cut is, in a very real sense, x, and that means x is a real. QED.

That needs tidying up and formalising, but it does work.

[lmm]

If you're using the Dedekind cut definition why use the line at all? Just say a real is any set of rationals bounded above, with arithmetic defined the obvious way; defining equality is slightly fiddly but it's fiddly with a number line too. What does the line visualization gain you?

[ColinWright]

Because it was asked how we knew we "got them all", referring to points on the line. The reals are a way of modelling the line, the line is a way of visualising the reals. Each is complementary to the other.

And besides, the rationals are totally ordered, and their completion is totally ordered, so it makes sense to think of them as arranged in a line. The problem is that the reals are very, very strange in some ways, and people do get seduced into thinking they understand them, whereas usually it's just a case that they've got used to them.