[gizmo686]

Do you have any idea of what set we should use to replace them with? The rational numbers can do a lot, but we have discovered that there are numbers worth talking about (and which can be described) that are not rational. Whatever replacement you propose must be usable where ever we would use real numbers, and must be at least as simple to use.

[moyix]

One possible replacement is the computable numbers [1]; this includes the algebraic numbers and some common transcendentals (e, pi), and you can even build up something akin to standard analysis (computable analysis [2]).

[1] http://en.wikipedia.org/wiki/Computable_number

[2] http://en.wikipedia.org/wiki/Computable_analysis

[gizmo686]

Unfourtuantly, there exist numbers which are definable but not computable.

[moyix]

Sure. Chaitin's Omega is a good example. The question is whether such numbers occur in the real world.

[jfarmer]

The trick here is defining "occur" and "real world" precisely. Are you saying the act of me writing down those symbols and expressing the idea does not count as "occurring in the real world?" ;)

[moyix]

This reminds me of the self-defeating property of an "uninteresting" number -- a reasonable definition might be "any number that does not have any property of human interest", but then of course there is a smallest such number, and so that has the interesting property of being the first uninteresting number, a contradiction!

You're right though that a more precise definition of "real-world numbers" is needed, but I confess that my attempts to think of one in the past few minutes have been essentially circular (coming down to "the ones we know how to compute")!

[jfarmer]

Well, we can and have made the idea of a computable number precise: http://en.wikipedia.org/wiki/Computable_number

It's not clear whether the universe is computable, however, in the sense that we only find computable numbers in nature. This is kind of an epistemological catch-22, though. How would we know whether this were the case or not?