Problems with axioms and their potential uses in real life.

Okay, so I am looking for more examples that revolve around this topic. My topic is: "Is the assumption of basic truth needed in order to create a system of mathematics that is reliable?" Here, when I reference basic truth I am talking about axiomatic systems. As of now I am going to talk about axioms in euclidean geometry and how they make sense because geometry can measure perfection. What else is there?

• When you say you are looking for examples, what are you looking for examples of? Also, could you clarify the last part: "As of now I am going to talk about axioms in euclidean geometry and how they make sense because geometry can measure perfection. What else is there?" I gather from this that you are looking for examples from Euclidean geometry, but apart from that I don't understand what you are saying. – Trevor Wilson Nov 14 '13 at 5:35
• @TrevorWilson I guess I am looking for examples where axioms are in play and very apparent. So in geometry they are apparent and make sense. In many other disciplines like Calculus ZFC is in play at all times. – Jeremy Correa Nov 14 '13 at 5:53
• I see. ZFC is the most important example then, in the sense that most mathematical theorems (when phrased appropriately) can be proved from the ZFC axioms. In many cases they can be proved from a weaker axiom system, however. Whether or not axioms are "necessary" in order to have a notion of proof is a different question. – Trevor Wilson Nov 14 '13 at 6:00