Note that your force grows unboundedly in N, so close to zero you have things that are arbitrarily heavy compared to their distance. So what this paradox really is about, is alternating series' that grow with N, and whether we can say that they add up to zero.
If we call the force between the first two bodies f12, then the series of internal forces on this system of bodies (using negative to denote vector component towards zero) looks like -f12+f12-f23+f23-f13+f13-f34..., where, again, each new term is bigger than the last.
If you split this sum up by interactions, it's (-f12+f12)+(-f23+f23)+(-f13+f13)..., so "obviously" it adds up to zero. But if you split this sum up by bodies, each term is negative (and growing!) so the sum must be negative infinity.
The typical physicist solution is to say that open sets aren't physical, and to get the best answer we should take the limit of compact sets.
If it's worth saying, but not worth its own post, then it goes here.
Notes for future OT posters:
1. Please add the 'open_thread' tag.
2. Check if there is an active Open Thread before posting a new one. (Immediately before; refresh the list-of-threads page before posting.)
3. Open Threads should start on Monday, and end on Sunday.
4. Unflag the two options "Notify me of new top level comments on this article" and "