I have read that an axiom is defined as "an obvious truth." I have also heard that an axiom is a truth so obvious that no proof could make it more clear. My question is: why is one thing considered an axiom and not requiring of a proof, but another thing might be called a Proposition and thus requiring a proof? Consider this the following example as an illustration of my point:
Axiom for the Order of Integers
If m and n are positive integers, then m+n and mn are positive integers.
Proposition
If m is an integer, then:
a. m ∈ Z+ iif m>0
b. m ∈ Z+ iif -m<0
What I don't get is why the proposition requires a proof but the axiom does not? Or another way, why isn't the proposition also an axiom? It seems just as obvious as the axiom and in fact the proof does not seem to make it more clear. If anything the opposite is true.
FYI: I am doing a first course in Discrete Math and this question is not for assignment or test purposes, just curiosity coming from having seen these definitions formally for the first time.
Basically, a set of axioms defines a mathematical structure. For example, the Peano axioms define what a natural number is, and the group axioms define what a group is. Propositions are true statements about the mathematical structure that can be derived from the axioms.
Now it may happen that different sets of axioms define the same mathematical structure, that is they are equivalent. Then an axiom of one axiom set may well be a proposition in the other.