[CogitoCogito]

> Only when you lose discreteness or compactness things start to get nasty. But this is just a flaw in our current definition of real numbers.

What "flaw" are you referring to? Also you're certainly free to use other definitions for real numbers, if you feel they better capture "reality".

[VyseofArcadia]

The problem I've always had with real numbers is that the vast majority (i.e. uncountably many) of them are not computable, and only countably many of them are computable.

That means that almost every real number, all but a vanishingly small subset, cannot be represented by any means whatsoever. No formulas, no algorithms, nothing. On top of which, they're surprisingly complicated to construct. There's several different ways of doing it, and they're all complicated.

At some point you've got to ask yourself, "are they really even there?" (Of course, those who already have non-Platonic leanings will find that question amusing.) I start to think that maybe we'd be better served by dropping all the non-computable numbers and start doing almost everything in the field of computables.

[enriquto]

All definitions of real numbers are equivalent, as far as I know. And as much as I love the definitions of R as a crowning achievement of human civilization, they lead to some infuriating paradoxes, especially in measure theory (e.g., freiling's axiom of symmetry).