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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?

user183748292
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    Everyday science, or even something like theoretical physics, is not conduced within a formal axiomatic system. Indeed, almost nothing is. – Dave L. Renfro Sep 12 '19 at 09:47
  • I mean only the mathematics which is shared by all – user183748292 Sep 12 '19 at 10:24
  • The same comment applies to "mathematics which is shared by all" (although I'm not sure what "all" refers to). Also, what about this situation? Assume Axiomatic System A is included within Axiomatic System B (for example, A could be ZF set theory and B could be ZFC set theory), and assume that a first person uses A and a second person uses B, both persons for tasks that only require A. Would we say that the second person uses A (because the tasks only require A) or would we say the second person uses B (because this is what the second person has available to use)? – Dave L. Renfro Sep 12 '19 at 10:48
  • This is what I was trying to ask. What are the main axiomatic systems used in the most basic approaches? e.g, when I study calculus, on which axiomatic system am I doing it? (maybe many possible? and if so how is the difference empressed in the subject? what is popular and what is not) I said the "shared by all" stuff only to emphasize why it is interesting, everyone who studies theoretical science is using mathematics, though I'm not sure how many know the base "rules". – user183748292 Sep 12 '19 at 11:00
  • The closest thing I can think of for what you're asking is the Reverse Mathematics program. For a relatively elementary exposition, see John Stillwell's 2018 book Reverse Mathematics: Proofs from the Inside Out (publisher's web page and amazon.com's web page). – Dave L. Renfro Sep 12 '19 at 12:13
  • Peano axioms for natural numbers. Axioms for real numbers and Axioms for sets : usually $\mathsf {ZFC}. Hilbert's axioms for geometry. – Mauro ALLEGRANZA Sep 16 '19 at 07:49

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