I am very interested in finding out about the foundations of math. When I feel foolish enough to try, I visit Wikipedia and end up in a truly endless rabbit hole. There are dozens of terms that I don't know: group, abelian group, field, structure, ring, algebraic structure, commutative ring, to name a very few. My background is very very shallow. I'm only now about to start taking my upper division classes for my bachelor's degree (so far I've only taken 2000 level classes such as calculus, linear algebra, discrete math, etc.) Could someone give me a very brief overview of how we start at the foundations and arrive to the algebra I know and love? I am looking for an explanation that starts from the very basics. Let's say I am an alien from a different universe. What is the first axiom you tell me, and how do you go from that to everyday math?