The question is not whether you can define it in terms of addition in some abstract way, but whether you can define it in terms of repeated addition, i.e. something of the form x + x + … + x.
> The question is not whether you can define it in terms of addition in some abstract way
This is splitting hairs. The reals themselves are defined "in some abstract way". For instance, what does addition of reals even mean? How do you add two arbitrary real numbers? Exactly, by adding the elements of the Cauchy sequences. (And similarly for multiplication.) This is the definition of addition (multiplication) and the only abstraction involved here is the abstraction due to the way the reals are constructed in the first place.
