The first half of (section one) doesn't comprise the foundations of abstract algebra in particular so much as the foundations of modern mathematics in general. The foundation of modern abstract algebra is more about axiomatic properties of structures like groups, rings, modules, fields, vector spaces, etc.
For that matter you actually don't need a lot of this machinery to derive analysis; for that all you need is a minimal subset (hah!) of set theory. Analysis focuses on the properties of continuous things, like real and complex sequences. You can derive the real and complex numbers axiomatically (like with Dedekind cuts of rationals) or synthetically (field axioms) as long as you have the definitions of sets, subsets, bounds, unions and intersections. I'm not even sure you need to care about the distinction between a subset and a proper subset.
That's not to say the material presented here isn't useful - it is. I'm just making the point that most analysis doesn't require abstract algebra aside from fields and (later on) vector spaces, and you don't need a whole lot to get there.
For that matter you actually don't need a lot of this machinery to derive analysis; for that all you need is a minimal subset (hah!) of set theory. Analysis focuses on the properties of continuous things, like real and complex sequences. You can derive the real and complex numbers axiomatically (like with Dedekind cuts of rationals) or synthetically (field axioms) as long as you have the definitions of sets, subsets, bounds, unions and intersections. I'm not even sure you need to care about the distinction between a subset and a proper subset.
That's not to say the material presented here isn't useful - it is. I'm just making the point that most analysis doesn't require abstract algebra aside from fields and (later on) vector spaces, and you don't need a whole lot to get there.