How do we decide which axioms are necessary?

I am studying the axioms for a complete ordered field. I have looked at different sources, some of which differ slightly in their listings.

Given some construction of the reals (e.g. Dedekind cuts or Cantor sets), it is relatively easy to show that any given axiom is satisfied, but what I can't quite get my head around is: how do we decide which axioms are necessary?

I do not mean "necessary" as in "minimal" (which is very easy to define: no axiom can be derived from the others). Rather, what I mean is probably a little more subjective. At the risk of offending someone on here, I tend to believe that 'applicability to the real world' has always been a (not the, lest I get assaulted) driving force behind mathematics, at least up to some point in time. Most people wanted the reals to be useful and convenient.

My question is: why are some axioms necessary for the reals to be useful, while others do not need to be stated?

I will give a few examples:

1. The way I justify the need for distributivity of multiplication over addition is that if we did not include it, we could end up with structures that do not conform to our experience (e.g. we could define the multiplication of two negatives as being positive). Is this a reasonable way to justify this axiom? Are there any more insights?

2. Multiplicative translation of $<$: $\forall x,y,z\in F, \, (x<y) \wedge (0<z) \implies xz < yz$. Of course we would like this property, but why is it OK to leave the case where $z<0$ undefined? Do the other axioms guarantee that we get what we 'want' in this case?

3. Without additive and multiplicative translation of $<$, can there be an ordering relation that satisfies the other axioms (trichotomy and transitivity) but violate additive and/or multiplicative translation?

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I think we would like a number system to be (1) applicable to the real world and (2) convenient. There is no difference between the rationals and the reals with respect to #1, since real-world measurements have finite precision. However, it's very convenient to be able to have $\sqrt{2}$ and $\pi$ and call them numbers. – Ben Crowell Mar 19 '14 at 5:39
@BenCrowell Agreed, and edited the question to reflect this. – MGA Mar 19 '14 at 5:50
You might want to consider checking out reverse mathematics. It studies this very subject: en.wikipedia.org/wiki/Reverse_mathematics – dezakin May 9 '14 at 23:29

Axiomatizations of things which have roots in observations of the real world tended to come late compared to the use of those things. Mathematicians used something like the real numbers for centuries before they were adequately axiomatized. With that background, the "necessary" axioms become "those things we always assumed before we went to the trouble of axiomatizing". So, when a historical object is axiomatized, the goal is to prove all the "obvious" things we've always assumed about them from a proposed set of axioms.

This is one of the reasons that axiomatization of set theory was so varied in the 20th c. -- there were a number of things we "knew" had to work and they didn't all do so in particular axiomatizations. (Should set theory contain a universal set? ... Axiom of (global) choice? ... Grothendieck universes? ZF contains none of these.)

With respect to your particular examples:

1. Distributivity of left multiplication over addition is involved in ensuring that the reals are not branched. In particular, the interval $[0,1)$ covers $[0,2)$ via $2[0,1)$ and $[0,1)+(1+[0,1))$ and these two choices give you the same thing (via the axioms). (This isn't well explained here. More in part 3, below, where avoiding the possibility of branching (via the above choice) is discussed.)

2. Let $z<0$, then $z+(-z)<0+(-z)$ or $0<-z$ and we get what we expect via additive translation.

3. Without the axioms you mention, there can be incomparable pairs. Consider the branches of a tree. Along any sequence of branch choices starting at the ground and proceeding continuously upward, it is easy to compare any two points on the path with $<$,$>$,and $=$. However, along two different choices of branching, it is not clear how to compare two points except that they're greater than the common initial segment of choices. They are otherwise incomparable. This is not "fixed" if the branches all eventually come back together -- in that case they are also less than the common final segment. The other two axioms ensure (in a roundabout way) that the real line is a single line and there are no branchings that can lead to incomparability. (Contrast with "poset" or "partially ordered set".)

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