Introduction to Mathematical Thinking
Coursera
Learn how to think the way mathematicians do – a powerful cognitive process developed over thousands of years. Mathematical thinking is not the same as doing mathematics – at least not as mathematics is typically presented in our school system. School math typically focuses on learning procedures to solve highly stereotyped problems. Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself. The key to success in school math is to learn to think inside-the-box. In contrast, a key feature of mathematical thinking is thinking outside-the-box – a valuable ability in today’s world. This course helps to develop that crucial way of thinking.
More resources on Axiom of Choice
MathWorld: Axiom of Choice
Concise reference with proofs
nLab: Axiom of Choice
Category-theoretic perspective, favored by practitioners
Stanford Encyclopedia of Philosophy: Axiom of Choice
Comprehensive philosophical and mathematical overview, top rec in r/math
Banach-Tarski Paradox and Axiom of Choice
Numberphile video explaining AC via Banach-Tarski, highly upvoted in r/math
Logic and Set Theory
This course is an introduction to Logic from a computational perspective. It shows how to encode information in the form of logical sentences; it shows how to reason with information in this form; and it provides an overview of logic technology and its applications - in mathematics, science, engineering, business, law, and so forth.
Set Theory
Learn the axiom of choice and its implications in set theory. Taught by William Hamilton.