How do we get the usually used form of $<$ (i.e. $\dots-3<-2<-1<0<1<2<3\dots$) on the real number field, defined as the Dedekind complete totally ordered field?
Why isn't it e.g. $\dots-3<-2<1<2<0<-1<4<5\dots$? $0$ is defined as the additive identity of $\mathbb{R}$ and $1$ as this set's multiplicative identity. I'm especially curious about why the additive and multiplicative identities are placed the way they are.
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What kind of properties do you want real numbers to have? For example, if you require $x^2 > 0$ for all $x \neq 0$, then you get $1 = 1^2 > 0$.
Looking a bit more deeply, in Dedekind's cut, you want to "fill the gap" between rationals, so to speak. Therefore, it will be preferable if the addition, subtraction, order relations etc of real numbers behave the same way they do for rationals. How will you decide the properties for rationals? Well, they come from natural numbers, which has a (tentative) basis in "counting", and we all "know" 2 is greater than 1!
If you want a bottoms up view of the whole construction, consult Terence Tao's Analysis I. It is an excellent read once you have some exposure to limits (it might tear down some preconceived notions, at least it did in my case. Boy, was that a learning experience!!!)
This is how I eventually made my peace with real numbers: we have certain expectations of their behavior, so what will you need to concretely define them and retain those expectations?
Bottom line (personal opinion), we don't construct reals, rather we construct axioms that describe real number system logically.
They aren't placed there. They were already there. Natural numbers were for counting. If you want to count backwards, you would need to extend to integers. If you want to count in halves or thirds and so on, you would need to extend to rationals. Finally, if you have a sequence of rationals that 'converge' (such as Cauchy sequence), you need real numbers otherwise the 'limits' would not exist. So where $0$ and $1$ end up in the real line is simply where they started off in the natural number sequence. They didn't choose to have particular locations in the real line, nor did we. All those intervening rationals and reals just 'happen' to fill the 'gap' in between $0$ and $1$.