Genady

Yes, I'm studying this book: This notation, (rather than, e.g., S or s) is new to me.

This note of yours has a direct relevance to this "Quiz" of mine here: https://www.scienceforums.net/topic/140398-from-naturals-to-integers-quiz/ 🙂

The construction that I've learned recently follows closely the "2.12. Schanuel (et al.)’s construction using approximate endomorphisms of Z ([2, 11, 16, 29, 30, 1985])" in your first linked paper. Interestingly, my book cites rather "Norbert A’Campo, A natural construction for the real numbers, Elemente der Mathematik, vol. 76 (2021)." P.S. Ah, I see that A'Campo's is your second linked paper. Perhaps, there is some difference that I didn't see yet.

Thank you! I didn't know about 10 different ones, only about three, I think. And they all constructed rational numbers before constructing reals. So, a direct route from Z to R without Q was interesting.

Real numbers can be constructed directly from integers, without a construction of rationals.

Given natural numbers [math]\mathcal{N}=\left\{0, 1, 2, ... \right\}[/math], Why do they identify [math]x \in \mathcal{N}[/math] with [math]\left\langle 0,y \right\rangle[/math] rather than [math]\left\langle x, 0 \right\rangle[/math]? Does it matter? If yes, how / when?

Right. As they say in the first part of the definition, xRy and x=y are mutually exclusive.

That's it!!

Ha, that's too. Still, there is another one, related.

Without going into any technical details, the short answer is,

No. (Check your notes or references about basics of algebra.) I've checked. Turned out that I am right. Can you find what is mistaken in the quoted definition? (Read it carefully in the OP.)

No topology here. This is pure set theory, more specifically, ZFC. There is no requirement that a set has to be well-ordered. It is only a definition, when it is. I claim that the definition as stated is meaningless, i.e., no set so well-ordered exists. (I don't remember which quiz/riddle on divisibility I posed months ago ☹️ ) P.S. I like your "Location." 🙂

Consider these definitions: I think, there is a mistake in this definition of well-ordering which makes the latter impossible. Am I right?

It appears that you have three unknowns: [math]x, y, \alpha[/math], and you have one equation [math]yx=\alpha[/math]. You need two more equations to solve the problem. If I understand your sketch correctly, one equation could be that two red areas are equal, and another that the brown area equals sum of the red and the green areas. If so, the rest is some trigonometry and some algebra.

There is a theorem in logic that says that given a countable formal language, any consistent formal theory in this language has a countable model. What is paradoxical about this statement? For example, the language of the axioms of Zermelo-Fraenkel Set Theory ZFC only consists of the membership relation ∈. Hence, if ZFC is consistent, it has a countable model. However, it is easy to prove from the axiom system ZFC that there exist uncountable sets. How is it not a contradiction?