Is it possible to develop Analysis solely from Peano's axioms

Question

...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?

Answer 1

It isn't true that "Peano's axioms are all we need". Landau's beautiful account involves some set theory (the Dedekind cuts are sets or pairs of sets of rational numbers). There are many elegant ways of getting to the real numbers from the integers, but they all need a bit more than Peano's postulates. One of my favourites is this paper by Behrend, which shows how the additive structure can be defined first and the multiplicative structure derived from it.

Answer 2

For what it is worth, I was able to construct the set of positive real numbers as Dedekind cuts starting from Peano's axioms and using the rules of first-order logic and some set theory (my own formulation based very loosely on ZFC).

I began with Peano's axioms as follows:

1. $1\in N$
2. $\forall x\in N:[ S(x)\in N]$
3. $\forall x,y\in N:[S(x)=S(y)\implies x=y] $
4. $\forall x\in N:[S(x)\ne 1]$
5. $\forall P\subset N: [1\in P \land \forall x\in P:[ S(x)\in P] \implies \forall x\in N:[x\in P]]$

I then constructed the usual addition ($+$) and multiplication ($\times$) functions on $N$, and constructed the set $Q^+$ of positive rational numbers such that:

$\forall x:[x\in Q^+ \iff x\subset N^2 \land \exists a,b\in N: \forall c,d:[(c,d)\in x \iff a\times d = b\times c ]]$

Note: By constructed, I do not simply mean defined. I mean that I actually prove the existence of these constructs using the axioms of set theory and the rules of logic.

I was eventually (after thousands of lines of formal proof) able to construct the usual addition ($+$) and multiplication ($\times$) functions, and the usual ordering ($<$) on $Q^+$, and the set $R^+$ of positive real numbers such that:

$\forall x:[x\in R^+\iff x\subset Q^+$

$ \land \exists y:[y\in x]$

$ \land \exists y \in Q^+:[ y\notin x]$

$\land \forall y\in x: \forall z\in Q^+:[z<y\implies z\in x]$

$ \land \forall y\in x:\exists z\in x: [y<z]] $

From set theory, I had to be able to construct subsets, power sets, Cartesian products (ordered pairs and triples only) and functions of one or two variables. I also made use of set equality (extensionality). I made no use of any other aspects of set theory.

Addendum

Here are links to to key proofs for constructing the positive real numbers:

• Constructing the addition function $+$ on $N$ (the positive integers) (727 lines)

• Constructing the ordering $<$ on $N$ (53 lines)

• Constructing the multiplication function * on $N$ (736 lines)

• Constructing the set of positive rational numbers $Q^+$ (91 lines)

• Constructing the ordering $<$ on $Q^+$ (51 lines)

• Constructing the set of positive real numbers $R^+$ (70 lines)