Are there in mathematics some non-logical axioms which are not obviously true but we accept them as true?

Question

I have encountered some non-logical axioms (e.g., a + b = b + a) in a linear algebra (field axioms, vector space axioms). For all those non-logical axioms I have encountered, it seems to me that they represent some obvious common sense truth. I wonder if there are some non-logical axioms in mathematics which we just accept as true without it being obvious or common sense.

Answer 1

In many cases, axioms don't say “this is obviously true” but “this is the type of structure we are talking about”.

For example, it is not obvious that every element has an inverse, but that is one of the group axioms, not because we can't think of cases where this is not true (we need to look no further than to the natural numbers for this), but because in group theory we are specifically interested in those cases where it is true.

If we don't care about every element having an inverse, we are talking about monoids, whose definition differs from the definition of groups precisely by not having the axiom that every element has an inverse.

Answer 2

When axioms were first proposed in ancient Greece, they were meant to be truths we don't dispute. Nowadays, axioms aren't required to be "true", whatever that means. We settle for asking what follows from a given choice of axioms, and choosing whichever consistent-as-far-as-we-know axiomatic systems are convenient for us to do a lot of mathematics. There are many such systems, but that's OK, because they're typically taken as definitions. @celtschk already gave the example of a group being defined by the axioms is satisfies.

In some cases, the definition is implicit, such as when we define sets as "the things our favourite set theory, e.g. ZFC, talks about". (If we'd chosen a set theory with urelements, sets would be defined to exclude those, of course.) This is a bit different from the case of groups, because I can't say, "this thing satisfies all ZFC axioms, therefore it's a set". What I can say is, "this structure satisfies all ZFC axioms if they're consistent, therefore it's a model of ZFC, and the things in it are sets". ZFC is interesting because we can use it as a foundation for most of modern mathematics, including all the mathematics we'd be interested in before studying set or category theory.