Well first consider how the natural numbers were constructed:

1 =

2 =

3 =

So each element of the new number is the set of one of the numbers that came before it. We also include the empty set as an element of each new number as well. (It's also how we get ordering, as seen 2 is a subset of 3)

So C is the sum of A,B if and only if the cardinality of C is equal to the trivial cardinality (equivalent elements are treated separately) of .

Just so you know I just came up with this... so if its flawed or just dumb... don't hate too hard