I read this in several books, and there's a Wikipedia article unquestionably stating that reals must be representable by means of regular language generated from finite alphabet.
My questions are:
- What would be generated from a regular language with an infinite alphabet?
- What are the devices we use to extend this alphabet (such as periodic fractions and continued fractions)? I.e. if finite alphabet with just the juxtaposition was enough to construct all reals, what are we constructing when we use periodic fraction notation?
- If we construct reals as Cauchy sequences, then every term in these sequences is unique and the sequences are infinite: then how come we find a bijection from infinite sequences generated from infinitely many members and sequences generated from finitely many members?
- Alternate objections: Cauchy sequences (when used in the construction of real numbers) are convergent sequences, which would imply that every real number, written as such a sequence, must converge to something, but isn't this a restatement of the conjecture about normal numbers? I.e. doesn't convergence require that reals must have statistically predictable number of all digits? Since I don't know of that being proved, wouldn't decimal expansion of reals still be a conjecture, not a proved fact?
I was trying to read this article: https://www.dpmms.cam.ac.uk/~wtg10/decimals.html which goes into more details about constructing reals in this way, but I don't feel satisfied with the paragraph where it talks about the least upper bound (which I know to be the distinguishing property of real number field, which rationals don't have). But I don't feel satisfied with a proof. It keeps saying "it's easy to see" and such, while I don't find it to be easy to see... If you are willing to lay out an answer, just elaborating that part of the article would be enough for me. For example in the mentioned paragraph, the author defines the supremum to be such and such decimal expansion, while I don't think this is a matter of definition. I think that one must prove existence instead. (What would have stopped me from defining a decimal expansion even closer to the extremum than the one the author of the article chose?)