I love Calculus, but I fail to see how Real Analysis is actually a way of proving anything. The whole subject seems completely arbitrary. Sure, it seems to make things that look and behave like the real numbers, but how do we know this isn't just a coincidence? Where do axioms come from in the first place? It seems like we could easily prove everything possible from whatever axioms we have, then forget our axioms, and then reprove our way back to our old axioms from our proven results. It just seems fishy to me, why should I believe any of it is actually true?

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kainui · Answer

For instance, if I take the intersection of an infinite number of open sets, I could end up with a closed set. Now why is it that I can't find a number that's less than another number with no numbers in between? Sure, you can add the two numbers up and divide by 2, but how do we know that as we approach infinitely close that things don't change, and that there really is a number next to another number with no numbers in between? It's all just sort of fake feeling overall.