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:
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?
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?
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?