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Manfred comments on Open thread, Jul. 17 - Jul. 23, 2017 - Less Wrong Discussion

1 Post author: MrMind 17 July 2017 08:15AM

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Comment author: Manfred 17 July 2017 08:05:50PM *  0 points [-]

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.