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]
Ok thanks all. Complex, I will try to familiarize myself with the subject before considering any book purchase, so I must put you on hold. Halls, I will check the link you gave as soon as I can.
