When we apply mathematical theorems to perform some computation or to prove some statement, we are relying on an axiomatic system. I am currently finishing computer science studies which include a lot of mathematics but I feel I lack of the origin of these theorems. I am interested in the questions:
What are the most common axiomatic systems in mathematics?
For example, what are all our assumptions when doing calculus (I mean in general)?
How much do they vary between ordinary subjects such as doing calculus or linear algebra? do they differ at all (If it is a matter of choice, then what are the popular ones)?
What are the main advantages and disadvantages in each popular axiomatic system?