What reading / mooc / online video etc would you recommend for an software engineer who wants to understand the reformulation of mathematics using set theory. My background is in machine learning, signal and image processing with good knowledge of Calculus, linear algebra, functional analysis, Fourier analysis and a taste for math :)
EDIT: What I know is that the movement started in the 19th century and into the 20th century with the reformulation of Calculus. The motivation was, I believe, to give Calculus a foundation based on Logic to avoid contradictions (I don't recall what the contradictions or paradoxes were).
Based on the text of your edit it sounds like you want to study rigorous analysis. I would recommend "A radical approach to real analysis" by Bressoud as it takes a historical approach and teaches the math along with the history of the overhaul in the 19th century you are alluding to. You might also appreciate the fact that he motivates things with the familiar problem of trigonometric series that was quite controversial in the 19th century.
I wouldn't characterize it as a 'reformulation using set theory' (although any treatment nowadays uses set notation and has some amount of set theory in it.) The rigorous approach to calculus actually came first and set theory was born out of it. (Cantor came upon the concept when he was studying the uniqueness of trigonometric series!)