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 of unpaired forces will grow until it peaks and then shrinks down to zero (as the number of forces that pair up will eventually overcome the added forces).
What happens if you take this approach with an infinite number of particles? Well, at each moment you add an infinite number of new forces, and only a finite number pair up. Thus, as soon as you start the calculations you have an infinite number of unpaired forces and that number only grows.
The right way to do this is the buddy system: never let a force enter the calculations without its reverse force. To do this, you add particles to your consideration one by one. The number of considered forces is N choose 2, and you never have unpaired forces.You let the number of considered particles grow to infinity, and it still works.
What happens if you take this approach with an infinite number of particles? Well, at each moment you add an infinite number of new forces, and only a finite number pair up. Thus, as soon as you start the calculations you have an infinite number of unpaired forces and that number only grows.
It certainly doesn't grow, it becomes infinity and stays there.
Okay, this is getting into set theory, but its actually the case that adding infinitely many things and then taking finitely many away can eventually leave you with zero.
Suppose at step n you add the foll...
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.