Which branch of math does that?

You may want to investigate abstract algebra.

 

I'm not sure exactly what you mean, as obviously nobody examines proofs of axioms!

 

Provable from what? If they are truly axioms then they can't really be proved true or false from other statements.

 

A logical assumption is very different from a proof.

 

If you are interested in a proof of the Peano axioms, you might want to look at an introductory set theory book, such as Halmos'.

 

Herstein's, Topics in Algebra, and then there are a few unknown books that I found which surveyed selections of abstract algebra, to give you an idea of how broad the spectrum is and provide you with an illustration so that you can select areas that you enjoy. If you are curious, I can give you those as well (although, they are REALLY old and seriously, no one has heard of them but I enjoyed pieces of them). I think this is the area that you are referring to? It's beautifully elegant, you will enjoy it.

As for axiomatic mathematics, I am under the impression that mathematicians construct axioms/assumptions/postulates to govern a particular formal logic system (such as an algebraic system) and from there, interesting properties emerge. If the system produces interesting properties, the system is studied further.

Mathematics are deductive because if the premises are true then the conclusion must be true because the conclusion is contained in the premises. However, the premises must be assumed true.

 

Given any system of axioms, there always exist a simpler system in which those axioms can be proved as theorems.

Peano's axioms assert that there exist a set of objects, N, called "natural numbers, and a function s (the "successor function"), from N to N such that
Axiom 1: There exist a unique member of N, call "0", such that s is a one to one function from N to N\{0}.
Axiom 2: If a subset, X, of N contains 0 and, whenever it contains n, it also contains s(n), then X= N.

You can, however, define the natural numbers in terms of sets: 0 is the empty set, 1 is the set containing only 1 (only the empty set), 2 is the set containing only 0 and 1, and, in general, given any n s(n) is the set containing n and all of its members. From that one can show that Peano's axioms are true.
Note: Historically, Peano's axioms included the number 0 as I have here. Nowadays, however, most people start with the number 1.

Werg22, you might find this interesting:
http://academic.gallaudet.edu/courses/MAT/MAT000Ivew.nsf/ID/918f9bc4dda7eb1c8525688700561c74/$file/NUMBERS.pdf [Broken]