Then you are calculating the force caused by N+1 particles on N particles. For every particle i, you look at the i-1 to the right, add up their gravitational force, and see that it is dwarfed by force from the particle to the left- particle number i+1.
If you have a finite number of particles, the mistake vanishes. If you have an infinite number of particles but you add particles all at once instead of half of i and half of i+1 at once, the mistake vanishes.
I calculated using all the particles to the left rather than just one, and so every pair got taken into account once for each member of that pair.
We've discussed Edward Nelson's beliefs and work before. Now, he claims to have a proof of a contradiction in Peano Arithmetic; which if correct is not that specific to PA but imports itself into much weaker systems. I'm skeptical of the proof but haven't had the time to look at it in detail. There seem to be two possible weakpoints in his approach. His approach is to construct a system Q_0^* which looks almost but not quite a fragment of PA and then show that PA both proves this system's consistency and proves its inconsistency.
First, he may be mis-applying the Hilbert-Ackermann theorem-when it applies is highly technical and can be subtle. I don't know enough to comment on that in detail. The second issue is that in trying to show that he can use finitary methods to show there's a contradiction in Q_0^* he may have proven something closer to Q_0^* being omega-inconsistent. Right now, I'm extremely skeptical of this result.
If anyone is going to find an actual contradiction in PA or ZFC it would probably be Nelson. There some clearly interesting material here such as using a formalization of the surprise examiation/unexpected hanging to get a new proof of of Godel's Second Incompleteness Theorem. The exact conditions which this version of Godel's theorem applies may be different from the conditions under which the standard theorem can be proven.