I don't know if this leads anywhere, but here are a few thoughts on the problem.

It seems helpful to try to get away from the numbers 15 and 7, and to see the problem in a more general setting.

Conjecture.Let S be a set of integers, containing at least two elements, satisfying the following properties:

(1)S ⊆ S + S,(2)if andthen |T| ≥ n.

Then.

Remarks: (a) Condition (1) just says that every element of S is the sum of two elements of S. (b) The case n=8 is equivalent to the given problem about 15 and 7.

To get a feel for the problem, I looked at the cases n=2 and n=3.

For n=2, condition (2) just says that 0∉S. In that case, it's easy to convince yourself that |S| has to be greater than 3, and that in fact |S|=4 is possible, for example if S={±1,±2}.

For n=3, condition (2) says that if x∈S then –x∉S. In this case, it takes a bit more effort to convince yourself that |S|=5 is not possible, but that |S|=6 is possible, for example if S={–6,–5,–3,1,2,4}.

That's as far as I got. When n≥4 things start to get complicated. As I said, I don't know if this gets you anywhere. But at least it gives you a framework in which to experiment, and maybe get some ideas for a more general argument.