Genady

I remember getting a question on my US citizenship exam (many-many years ago), "What is Constitution?" with one line for an answer. I've answered, correctly, "Constitution is the supreme law of the land."

This notation, (rather than, e.g., S or s) is new to me. Only two days ago it was new to me, and it is already in my next book:

I've read it only last month and they've made a movie already 🙂

A good math joke, albeit 30 years old:

Right. And my book says, but still, since all the numbers here are integer, the definitions [math]|\lambda(m+n)-(\lambda(m)+\lambda(n))| < M_{\lambda}[/math] and [math]\left\{ \lambda(m+n)-(\lambda(m)+\lambda(n)) \right\} \,\text{is finite}[/math] are equivalent.

I still don't see a difference between the two constructions mentioned above. The first says, The second, Does anybody see how they are different?

Whoever it was that downvoted you, I've balanced it by an upvote, just as you've done for me earlier.

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!!