That's what I can't understand... For example, I can easily write a program that produces some digits one by one, indefinitely (excluding mechanical failures etc). I won't ever see the whole sequence – as it never ends – but to my mind, it seems like the program describes it. (I think that's basically the computable reals?)

Sure,there are a large number of "computable reals" such as square roots and taylor series expansions (pi, e, sine, cosine, etc.) which can be estimated to greater and greater precision (always with rational numbers) so we can reasonably talk about them in the abstract (such as use them in equations) much in the same way we talk about the imaginary numbers, quarternions, and octonions. But the idea that you need to construct an entire field of completely undefinable infinite chains of decimals just to allow for a small handful of irrational numbers is kind of absurd.

Norman Wildberger (among others) has put a lot of effort into building up the foundations of mathematics using only the rational numbers and it is quite interesting how far he has gone. He's considered to be a crank by many, but you have to appreciate his effort.

To go back to your original question, his Math Foundations video series, especially #14, #95, and #96 might help.

> Well, you can describe a negative number or a rational number, but an infinite sequence of non-repeating decimals is a different matter.

I'm a bit confused... it sounded like you had objections to defining numbers with infinite decimal expansions, but you're okay with e, pi, etc? Or did you just object to uncomputable reals?

Also I don't know much about this whole topic, but I was taught that you can't fill a continuous line with just the rational numbers – you need the "undefinable" ones for that... What do you think about this?

Also thanks for the references, I might check them out to see what Wildeberger has to say.