...and a few definitions on the way?
When I studied Calculus using Spivak's book It was clearly shown that, in order to prove some fundamental theorems (intermediate value theorem being one of them), one had to assume an additional property of numbers (namely the least upper bound property). Other postulated properties included the field axioms and 3 more axioms (needed for the inequalities).
But then I skimmed over "Grundlagen der Analysis" by Landau (which was in the recommended reading by the way) and he developed the reals using Dedekind cuts assuming only Peano's axioms - the way I saw it. I also read on Wikipedia that the set of Dedekind cuts has the least upper bound property. So it seems possible that we can get from Peano's axioms all the way to theorems in analysis (or am I missing some additional assumptions?)
If it is true that Peano's axioms are all we need, then I want to know how far we can take them...in general (not just real analysis). Also is there an elegant way to prove the field axioms for real numbers from Peano's axioms?