>>12441413that's a weird essay from my point of view:
the claim that "R is the biggest jump in sophistication" is also very subjective, throughout my education i've seen people baffled by fractions or complex numbers rather than R
to me, "IVT and real numbers" is an even more basic concept than "roots of numbers"
when you first encountered how did you think about it? well i'm sure most people do one of the following:
1) they have intuition about "continuity and IVT" and they understand here that there's gonna be a number that gives 2 when squared
2) they think "well whatever this is, it's somewhere between 1.4 and 1.5, it's also between 1.41 and 1.42, and so on and so on, so it's somewhere there on the fucking line", now you could argue that this is intuition about "infinite decimal expansion" or about "taking a limit of an increasing/decreasing sequence" which is very close conceptually "supremum/infimum of a set"
i'm confident you could give an axiomatic definition of "real numbers" to a high schooler based on the description "real numbers contain the rational numbers, there's the order < on reals, and bounded sets have supremums"
this is how you attain rigor with regards to real numbers
Why does N exists? is it because the axiom of set theory tells you the set exists? no, the axiom of infinity was invented, because we wanted set theory to conform to our intuition
why does Q exists? because you constructed it out of N? no, it's the other way around: when you are an undergrad studying set theory, the fact that you can build "stuff that looks like rational numbers" gives us confidence that the language of set theory is expressive enough to include the math we knew so far
does Z exist because you constructed it out of N? no, same argument as above
and just like Q and Z, the construction of R in terms of dedekind cuts or cauchy sequences serves as an argument that set theory makes sense; NOT THE OTHER WAY AROUND
