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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?

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  • $\begingroup$ 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. $\endgroup$ – Trevor Wilson Nov 14 '13 at 5:35
  • $\begingroup$ @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. $\endgroup$ – Jeremy Correa Nov 14 '13 at 5:53
  • $\begingroup$ 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. $\endgroup$ – Trevor Wilson Nov 14 '13 at 6:00
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(If I understood the question correctly:) Essentially, yes. You must make some unverifiable assumptions - i.e. assume some 'basic truths'. This is a big topic in the philosophy of mathematics - the most relevant reading on the subject would be on Goedel's Incompleteness Theorems.

You may also find the idea of the Muenchhausen Trilemma relevant, which deals succinctly with what you seem to be finding difficult to accept: it is practically impossible to support every axiom with a proof - to do so would require either circularity or an infinite regress.

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  • $\begingroup$ I'm not sure whether Goedel's Incompleteness Theorems are relevant to the question, because even if we had been able to prove the consistency of all mathematical theories from some weak set of axioms (say PA) this still would not mean that axioms were not necessary for proofs. $\endgroup$ – Trevor Wilson Nov 14 '13 at 6:04

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