What exactly are axioms?

Question

The title says it all. What exactly are axioms? I mean, I know that is a statement which do not need to be proven. But what are the requirements to be an axiom? For instance, may I state something that seems to be false as an axiom? May I state an axiom that is already a conclusion of another theorem or conclusion?

Are there any books on how to build axiomatic systems and explains this "philosophic" part of mathematics? Is Russell's a nice option for this?

Thanks

EDIT

Thanks for such many responses. I am asking this because recently, I have read an "article" in which the author tries to use the non-aggression principle as an axiom, like mathematicians and scientists do, to prove some things like mathematicians do: Establish the axioms and prove conclusions. But this principle, in my view, seems not be true, since aggression does exist.

I guess this question should be redirected to some exchange on Philosophy...

Answer 1

See Schlimm - Axioms in Mathematical Practice

This paper has a nice philosophical discussion of axioms.

Answer 2

Axioms are nothing more than the premises of your specific problem that are justified by the meanings of their predicates. The axioms should not conflict with the other axioms you may assume. Therefore, if they are false in your axiomatic system, they cannot be added as axioms (premises) to your problem. The propositions that are conclusions of the other theorems are theorems, but not axioms.

Refer to 2011 - Barker-Plummer - Language, Proof, and Logic.

Answer 3

It's fine to use some non-aggression principle as an axiom and prove all kinds of things from that. However, to what extent the proven theorems apply to our reality depends on the extent to which the axioms apply to our world.

This is how it always is with mathematics. Whatever we prove in mathematics will be true for any world to which the axioms apply. That may include our world, but it need not be. We can prove all kinds of things using the axioms of Euclidean geometry, for example, but if our world turns out to be non-Euclidean, then the theorems need not reflect the state of affairs in our world.

Note, though, that even if our axioms don't perfectly describe our world, they may still be a pretty good approximation. Hence, any theorems we derive from them may still be 'close enough' to our reality to learn something from.