I'm trying to get an overarching understanding of the components of mathematical systems so that in my self study of each category of math I can break them down by their unique aspects, i.e. the operators they use, the major concepts they deal with (i.e. how calculus is about "change"), etc.
As far as my experience with formal math terminology goes, im rather weak, and i get utterly confused by the technicality required in formal definitions.
As a good starting point, I'd like to better understand what the difference is between an axiom, a theorem and a postulate. At my current level of knowledge i would use them interchangeably (lol), however I'm sure one is founded upon the others.
If someone could explain the logical hierarchy/relation between these three it would be greatly appreciated.
Axioms are the things that are taken as basic, unchallenged assumptions. Depending on what area of mathematics you are working within, these may change. For instance, all of the elementary results of arithmetic are usually taken as unstated axioms in higher branches of mathematics (so you don't prove $1+1=2$ in a calculus course, but might in a certain other courses).
Theorems are conclusions that can be drawn from a set of axioms by using the rules of logic. Not every such conclusion is bestowed with the title of theorem though: usually the conclusion has to be meritorious, and often the verification is non-trivial.
I've always thought of postulates as being slightly less fundamental than axioms, but nevertheless similar in their nature as assumptions from which other statements are to be proven. To give a natural language analogy: I take as an axiom is that I exist, and I postulate that my senses aren't being manipulated as part of a perfidious plot.