Note that each pair has to be taken into account twice.
Once for each member of the pair = twice
Then you were calculating the gravitational effect of an infinite number of external particles on a finite number of particles, which makes things worse.
No, I calculated it for every particle, and therefore I calculated it on an infinite number of particles. Obviously my calculations only considered one particle at any time, but the same could be said of calculations in problems involving finitely many particles.
Suppose you have a system of N+1 particles.
You calculate the gravitational force on the first particle. There are N forces on it; you now have N unpaired forces.
Now you calculate the gravitational force on the second particle. There are N forces on it; you now have 2N-2 unpaired forces (you added N more forces, but one of the new ones and one of the old ones are paired).
Then you calculate the gravitational force on the third particle. There are N forces on it; you now have 3N-6 unpaired forces (you added N more forces, and 3 choose 2 are paired).
The number...
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.