Terry Tao says:
In his wonderful article, Bill Thurston describes (among other topics) how one’s understanding of given concept in mathematics (such as that of the derivative) can be vastly enriched by viewing it simultaneously from many subtly different perspectives; in the case of the derivative, he gives seven standard such perspectives (infinitesimal, symbolic, logical, geometric, rate, approximation, microscopic) and then mentions a much later perspective in the sequence (as describing a flat connection for a graph).
The purpose of these notes is to do the exact opposite: We give a minimal and opinionated formal construction of calculus.
- Construction of the real numbers
- Real topology
- Measure (failure occurs here)
Goal: Construct the integrable functions via a process of three Cauchy completions: First of the rationals under absolute value, then of the finite unions of intervals under naive length measure, and then of the simple functions under naive integral. This is more efficient than the standard textbook approach. The axioms of choice, regularity, and replacement are not needed; we will only need 6 axioms from ZFC.
Current status: Succeeded until running into an obstruction while proving that our construction of Lebesgue measure is continuous. (It turns out that according to the literature, hitting an obstruction at this point is inevitable without the axiom of choice.) To make further progress requires an abstraction like “codable sequences of codable Borel sets”, which seems quite interesting, but will take big energy to properly understand and develop. So we gave up for now.
Last, and least/most importantly: