Addition in $\mathbb{R}$ preserves order

Question

By Zermelo's theorem we have the set of real numbers can be well-ordered. Now consider real number $a$ being the $i$th smallest number, $b$ being the $j$th smallest number. Is it true that $a+b$ is the $(i+j)$th smallest number? Thank you.

Answer 1

No. Let $a$ be the 3rd smallest and $b$ the 5th smallest number, then what about $c:=a-b$? If it is the $k$th smallest number, you want $b+c$ to be the $(k+5)$th smallest number, but that makes $k+5=3$ ...

Answer 2

There are many different well-orders of the real numbers. Some of them have the property you specify.

Begin the well-ordering with $1,2,3,\dots$ in order; after that add all the other reals to complete the well-ordering. If we do it that way, then the $i$th smallest (in the well-order) is $i$ itself, so of course the addition works for these.

Of course most real numbers are of the form "$k$th smallest" for no finite $k$.