elementary set theory - How to define the conception of a sum without the operation of addition?

In short: I look for a definition of a sum of any number of natural numbers in the terms of pure set theory. Until now, neither have I found such a definition in books, nor invented it by myself.

In details:

Let there be $n$ piles of apples on a table (${n}\in\mathbb{N}_{>0}$). Let $x_i$ be the number of apples in each pile (${x_i}\in\mathbb{N}_{>0}$, ${i}=1…n$). How to define the conception of “total number of apples on the table” through ${x_i}$, without using the operation of arithmetic addition?

All sources known to me reduce this conception to the arithmetic addition one way or another. But it seems not quite correct to me: addition doesn’t reflect the main point of the conception, but it only is one of the possible operations for calculating this “total number”. Besides that, the entity of “total number of apples on the table” exists regardless of the fact whether we perform any operations to calculate it.

Furthermore, addition is defined for two or more addends, while “total number of apples on the table” exists and is computable even if $n=1$.

I am interested in a definition in terms of pure set theory. Individual natural numbers (for example, $n$ and each of ${x_i}$) can be defined, e.g. as finite ordinals. I look for a definition of “total number” also in the context of set theory (e.g. as a result of unions, intersections and other set operations).

Is this possible?