From the Principle of Sufficient Reason to causal finitism

Causal finitism says that nothing can have infinitely many things in its causal history. Suppose causal finitism is false. Then surely the following scenario is possible:

1. There has always been a lamp.
2. There is a backwards infinite sequence of flips of the lamp switch at times t₋₁>t₋₂>t₋₃>....
3. Each time the switch was flipped, the lamp state flipped between on and off.
4. All lamp state changes are explained by switch flips, and are not overdetermined by any other causes.

Call this story the "Reversed Thomson's Lamp" (RTL). If RTL holds, the Principle of Sufficient Reason is false. To see this, suppose that the lamp is on right after t₋₁. Then it's on right after t_−n for all odd n. But there is no explanation why it's on right after t_−n for all odd n; it could have instead been on right after t_−n for all even n. It would be circular to explain its being on after the odd-numbered times in terms of its being off after the even-numbered ones, since by the same token its being off after the even-numbered ones would be explained by its being on after the odd-numbered ones. If one had overdetermination, one could explain things in terms of the overdetermining causes, but that would go against (4).[note 1]

So if the Principle of Sufficient Reason is necessarily true, RTL is impossible, and hence causal finitism is true.