Ask HN: Learn Math the Hard Way

Yes, all or nearly all! That the trig functions are the source of the most important orthogonal basis in Hilbert space theory need not be good to know just for physics or signal processing!

Let's see: Also I mentioned characteristic functions and Bochner's theorem which are core probability and not just physics or signal processing. Characterization of time invariant linear systems might be in mechanical engineering and might be tried even in economics! Seismic data analysis via the fast Fourier transform is, yes, in signal processing but is also mostly regarded as geology or just looking for oil or anything 'down there'.

I tried, as you can see, I really tried, to show how trig was for more than just your father's topics in physics and signal processing. I tried!

The proofs for the hyperreals involve a lot more machinery than the proofs for the standard reals. That is my educated opinion based on knowing both sets of proofs and constructions. (Of course this need not make infinitesmals a pedagogical disaster - very few students actually care much about learning the proofs.)

As for the model theory approach, I am intimately familiar with the ultrafilter construction. It uses choice. I know there is a second construction which I am not familiar with, but from what I've read it also requires choice. Both involve model theory. That's a mighty big sledgehammer for a pretty small fly.

Incidentally Dedekind cuts can be understood as follows. The set of reals can be equated with the set of points where you can cut the rationals into two. More precisely if X is a nonempty subset of the rationals with an upper bound, we get a cut of the rationals into the set of upper bounds of X, and things that are not an upper bound of X. Any two subsets can be considered equivalent if their set of upper bounds is identical. An equivalence class of subsets is a real number.

For any cut you can generate a unique set A of rationals that are not upper bounds, and a set B of rationals that are upper bounds. When you do this, all rationals in A are less than all rationals in B, and A does not contain an upper bound.

Conversely if we have a partition of the rationals into 2 non-empty sets A and B such that all members of A are less than all members of B, and A does not contain an upper bound. Then we have a cut of the rationals.

So there is a 1-1 correspondence of reals to places we can cut the rationals to partitions of rationals with that property.

Those partitions of the rationals are called Dedekind cuts.

(This is one of two constructions of the real numbers. The other, Cauchy sequences, turns out to generalize more usefully in the field of topology.)

I did find out that there is a constructive approach though, so the axiom of choice is not actually necessary. www.math.ucla.edu/~asl/bsl/0403/0403-001.ps