Spivak Chapter 1 Assumptions In working through chapter 1 of Spivak's Calculus, I'm confused as to what operations are defined and available for my use in proofs.  For example, Spivak defines subtraction in terms of addition, using the additive inverse.  However, he never defines addition!
What does $a + b$ mean, or $a \cdot b$, or $a^b$?  Ordinarily I would consider commutativity a property of the addition operation, but in chapter 1 of Spivak it's not assumed: if you use commutativity in a proof, it should be broken out as an explicit step.
What operators am I allowed to use, and what do they mean?  The problems at the end of chapter 1 are trivial unless you return to first principles, but what are those first principles?  If we aren't starting from set theory and the Peano axioms, where are we starting from?
 A: You don't need axiomatic set theory to get into Spivak at all; you start from what we'd call a set, the numbers, but it is not necessary to think about them in terms of sets that might represent them, or whatever: they're never considered to be anything other than black-box-ish objects in a set theoretic sense, and because of this and the overall lack of set nesting, naïve set theory is sufficient.
The Peano axioms are embodied within the given axioms.  Specifically, the existence of $0$ and $1$, and their behavior when considered with addition and multiplication, make $S(0)=1$ the only choice.
The definitions of addition and multiplication are literally what are described here: they are operations $(F,F) \rightarrow F$ that follow the given rules.  We are then free to generate elements as we please; the result, however, is inevitable: we have created the real numbers (well, technically, the rationals.  The reals require some more gyrations, which are in the back of the book).  This isn't a coincidence!  The field axioms are designed to make systems that act like numbers.  All this is covered in chapter 2.
The only thing now missing is exponentiation, and this is a mere definition: $$a^b = \underbrace{a\times a \times a \times \cdots \times a \times a}_{b\text{ copies of }a}$$
Later on of course you'll need something better to deal with non-natural $b$, but it will suffice for now, and by the time you get there, Spivak will be ready to show you the way.  It's in the chapter on inverse functions.
