[gweinberg]

The problem with "points on a number line" as a definition for real numbers is that it's not clear how you can tell if you have all of them. You can populate a number line as densely as you care to using just rational numbers, but that's not all of them, you're missing out on numbers like the square root of two. You can toss in the non-intergral powers of rational numbers, but you still won't have all of them, you're missing out on col numbers like pi (or tau, if you prefer). Even after you toss in every solution to every differential equation you can name, and every number you can generate using well defined finite or infinite serieses, there's probably some horrible diagonaliztion proof that says you still don't have all of them.

[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.

[velis_vel]

The set of numbers that can be uniquely defined in the English language in a finite number of letters is a countable set, because the set of finite sequences of English letters is countable.

[gweinberg]

That's what I would think. But since the set of reals is clearly uncountable, it seems to me that the precise membership of the real numbers cannot be unambiguously defined. There must be uncountably many reals that are not the solution of any equation that can be made using a finite number of characters. But if a "real" number cannot be specified, in what sense does the number exist?

[velis_vel]

If you want to take a constructivist viewpoint, it doesn't exist. You can define constructible analysis, where you only work with numbers that you can approximate arbitrarily well using a Turing machine (this is a subset of all numbers that you can define, since you can do tricks with the halting problem). But constructible numbers still don't have decidable equality, since the halting problem reduces to constructible equality: is the number whose 2^-ith place is 1 if and only if the Turing machine M halts on the ith step equal to 0? You can approximate it arbitrarily well by running M for more and more steps, but proving that it's 0 would require proving that M never halts. (You can, however, get decidable ordering if you know a priori two numbers are unequal, simply by approximating them close enough that you can distinguish them.)

Personally, I'm not a constructivist; I think that these undefinable real numbers exist just as well as the ones that we can define. But that's a philosophical argument and I was never any good at those.