Alice has lived forever in a universe with an infinite road that has a beginning and no end, and is marked every mile. Every day of her life, by an irresistable longing, she has followed these rules:
If somehow she’s not on the road, she goes to mile zero on the road.
If she is on the road, she walks a mile in the endless direction of the road.
Besides the movement required by 1 and 2, she stays in place.
Where is Alice now? Nowhere! There is no possible present scenario compatible with the rules. She can’t either be at mile zero or off the road, because if two days ago she was on the road, she would now be on the road now be past mile zero, and if two days ago she wasn’t on the road, she would now be on the road past mile zero. She can’t be at mile n, because then she would have to have been off the road some time back, which violates the previous argument.
The rules are all coherent. A person who has to follow these rules seems to be possible. Yet the story is impossible. What went wrong? The neatest explanation seems to be that a (causally connected) infinite past is impossible.
(This is inspired by a recent infinite past Grim Reaper story I read from Rob Koons.)
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