[soVeryTired]

By no means am I an expert in this stuff, but don't you need the axiom of choice (or maybe something just a little bit weaker) to construct the reals?

I don't think it's fair to say the reals 'most certainly exist' without being misleading to a layman. They exist given some axioms that are used overwhelmingly often in mathematics, but you can still do some interesting stuff without those axioms, or with their negation.

[osoba]

You're correct. There is one object in all of mathematics that isn't defined (you've got to start from somewhere), it is just assumed that humans implicitly understand this concept, and it's the concept of a set (denoting a collection of objects).

Along with it, certain properties of this object are assumed (among them the ability to choose an element of a non empty set - this is typically called the axiom of choice). For a full list you can take a look at http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html

[soVeryTired]

But the axiom of choice is logically independent of the other zermelo frankel axioms. You can assume zermelo franked plus choice, and get the standard framework (ZFC) that let's you build the reals.

But you can also assume ZF plus the negation of the axiom of choice, and get a system that is consistent if and only if ZF is consistent. It's not clear (to me, at least) that this other system will let you build the real numbers.