[ginnungagap]

One can have a function in memory when the real number is nice in some sense, your example is algebraic.

But what if I want to represent an uncomputable number?

Or regardless of that, under any reasonable encoding of programs that can be held in memory by a computer, there are only countably many programs.

[jostylr]

We also cannot represent all natural numbers or rational numbers in a computer. But the ones we care about, we generally can. I guess one question is whether there are uncomputable numbers that we need to compute for some purpose other than just computing it as a challenge? And if there are such things, how is it usually done? The theoretical definition of an oracle is not problematized by being uncomputable by Turing machines, but it is in uncomfortable tension with the driving purpose of the definition. I think that is a bonus. I think we should pause to consider the relevance of numbers that are uncomputable.

There are a number of real numbers that one can define which can depend on whether we can prove something or not. It may turn out that we can never prove it and the number is never resolved.

My attitude, which may not be satisfactory, is that we do what we can and we should have a language/framework for facilitating that. I think the oracle approach highlights what we know and marks what we can't compute clearly. I call it the resolution of the oracle. I don't want known precision to be lacking just because of a poor definition of what a real number is.

As for the example being algebraic, it is a particularly nice example and it is the same example in every example of a Dedekind cut. Another example of such a rule, one which is not algebraic, would be whether pi is in an interval or not. Given a<b, one rule could be that it is No if b < 3 or a > 4 or sin(a)sin(b) > 0. Alternatively, it would say Yes if a >= 3 and b<= 4 and sin(a)sin(b) <= 0. To compute this, one needs to compute sine of a and b sufficiently precisely to determine their sign.

The flavor I am trying to convey is having a definition convey a useful goal. The interval approach says that we are trying to generate precision about our inaccuracy. I think this is something which would greatly benefit those learning about real numbers. Most of the time, the error is presented as secondary and an annoyance, the concerns of the error propagation in further computations is pushed to the side, and it is all relegated to experts or computers. The expansionary nature of arithmetic in error propagation is pushed to a usually unsatisfactory discussion about significant digits.

My goal here is to change the mental framework so that these concerns come to the forefront. Ideally, they also come with useful tools to handle the uncertainties such as ways to compute how narrow the input intervals need to be when doing a computation. And maybe, just maybe, students could become more comfortable with fractions.