> Or the axiom of choice: "the product of a collection of non-empty sets is non-empty." What a surprise. Right. Since it's obvious, the Banach-Tarski "Paradox"

> Or the axiom of choice: "the product of a collection of non-empty sets is non-empty." What a surprise. Right. Since it's obvious, the Banach-Tarski "Paradox" must be a trivial corollary and intuitively obvious, correct? http://mathworld.wolfram.com/Banach-TarskiParadox.html http://mathworld.wolfram.com/Banach-TarskiParadox.html Also: Any two objects in three-dimensional space which do not extend to infinity and each containing a ball of arbitrary size can be dissected into each other. Perfectly obvious, right? This follows from Banach-Tarski, and therefore the Axiom of Choice. A fuller development of the Axiom of Choice: http://plato.stanford.edu/entries/axiom-choice/ http://plato.stanford.edu/entries/axiom-choice/ Some implications of the Axiom of Choice, and of not taking it: http://mathoverflow.net/questions/20882/most-unintuitive-application-of-the-axiom-of-choice http://mathoverflow.net/questions/20882/most-unintuitive-app... My point is that mathematics is all about taking obvious statements, such as the definition of sets and set cardinality, and using them to prove non-obvious results, such as the fact there are as many rationals as there are integers, and, in fact, there are as many rationals strictly between 0 and 1 as there are integers total. The very best mathematical proofs are magicians, taking ordinary household items, waving and knocking a few times, and pulling a rabbit out of their cane in a way that is eminently fair but that you still didn't see coming. So, yes, an axiom by itself is perfectly obvious. It's just a prop. The wonder is in how the props get used.