Unfortunately I still have a soft spot for beautiful writing. The "point" often sticks better when it is expressed eloquently. Having to think a little bit to get to the point also helps with absorbing it, at least in my experience.
> I am slowly replacing the Cotton/Hazlitt translation with a contemporary one and adding new notes
So I would assume that the essay you're talking about is from the earlier Cotton translation and has still not been replaced.
This is the first time I've seen AI being used to "modernize" old texts, and it works wonderfully in this case; though a bit of the old-timey charm is lost imo. I used to read a translation that I'd found in my university library which I enjoyed a lot. Very readable but still retained the "feel" of a 16th century book. I don't recall the translator unfortunately.
Everything you are looking for is provided in the Epilogue of Spivak. For instance, in chapter 28, you will be able to show that 0a = 0 in any field. In chapter 29, you will rigorously define the reals as well as the operations on them. You will also show that they form a complete ordered field, which allows you to use the results from chapter 28. In chapter 30 you will show that all complete ordered fields are "essentialy the same" as the real numbers---i.e., the real numbers are "unique."
At the moment I'd advise not worrying much about the construction of the reals. Ideas such as limits, continuity, differentiation, integration, and even fields are much more important for later mathematics and applications (abstract algebra, topology, geometry, physics) than the construction of the reals. Constructing the reals is pretty much something you do a couple of times (traditionally once with Dedkind cuts, as in Spivak, and once more with Cauchy sequences) and then never think about again.
Edit: I'm not sure what you mean by "something I could code." If you want something that you could type in a proof assistant you might have some luck looking at the mathlib library of Lean https://leanprover-community.github.io/mathlib4_docs/Mathlib....
HN has been surprisingly good today. Seen a couple of very high quality posts (at leasts posts that are to my liking), this one included. I thank OP for posting this.
The kinds of AP Calc kids you see must be super different from the ones I've seen then. I'm in a uni with a large amount of kids who've taken AP Calc and yet most of them struggled a lot with the first semester intro-to-proofs course. The second semester linear algebra course (it's a middle ground between a proof-based and a computational course) was even worse. I know many kids with 5's in AP Calc BC who resort to memorizing basic proofs (which is the sort of thing that helps with AP exams) instead of learning how to write one on their own, and some of the TA's have told me that the most common mistake on the midterm was incorrectly negating the statement "A is a subspace of B".
This is not to say that high schoolers can't do abstract algebra (or higher mathematics more generally). In senior year I attended a week-long camp (Arnold had a full semester) in my local uni where we proved the impossibility of squaring the circle, doubling the cube, etc using field extensions. And I was in the older side! Most of the kids there were 10th grades. Though Arnold's class was probably substantially harder than ours. I worked through V. B. Alekseev's book after the camp was over and the exercises were substantially harder than the ones we did at the camp. The material on Riemann surfaces was very hard to understand as well, much harder than the group theory part (I still don't understand Riemann surfaces lol).
In conclusion, AP Calc, and students' performance in it, is a terrible metric for assessing mathematical ability. Sorry for the long rant.
I found out about this a few months ago through Cristobal's blog: https://cristobal.space/. Somehow didn't notice how the post mentions Omar's involvement at the top lol. Thanks anyway tho.
Do you know anywhere where one can look into dynamicland more deeply? I've been interested in playing around with it for a while (hopefully I can get my hands on a projector lol) but have never found any details. Omar Rizwan's website had a cool post on geokit but that was all I managed to find.