Satan’s Apple: Satan has cut a delicious apple into infinitely many pieces. Eve can take as many pieces as she likes, but if she takes infinitely many pieces she will be kicked out of paradise and this will outweigh the apple. For any finite number i, it seems like she should take that apple piece, but then she will end up taking infinitely many pieces.
Proposed solution for finite Eves (also a solution to Trumped, for finite Trumps who can’t count to surreal numbers):
After having eaten n pieces, Eve’s decision isn’t between eating n pieces and eating n+1 pieces, it’s between eating n pieces and whatever will happen if she eats the n+1st piece. If Eve knows that the future Eve will be following the strategy “always eat the next apple piece”, then it’s a bad decision to eat the n+1st piece (since it will lead to getting kicked out of paradise).
So what strategy should Eve follow? Consider the problem of programming a strategy that an Eve-bot will follow. In this case, the best strategy is the strategy that will lead to the largest amount of finite pieces being eaten. What this strategy is depends on the hardware, but if the hardware is finite, then there exists such a strategy (perhaps count the number of pieces and stop when you reach N, for the largest N you can store and compare with). Generalising to (finite) humans, the best strategy is the strategy that results in the largest amount of finite pieces eaten, among all strategies that a human can precommit to.
Of course, if we allow infinite hardware, then the problem is back again. But that’s at least not a problem that I’ll ever encounter, since I’m running on finite hardware.
We can definitely solve this problem for real agents, but the reason why I find this problem so perplexing is because of the boundary issue that it highlights. Imagine that we have an actual infinite number of people. Color all the finite placed people red and the non-finite placed people blue. Everyone one t the right of a red person should be red and everyone one to the left of blue person should be blue. So what does the boundary look like? Sure we can’t finitely transverse from the start to the infinite numbers, but that doesn’t affect the intuition that the boundary should still be there somewhere. And this makes me question whether the notion of an actual infinity is coherent (I really don’t know).
My best guess about how to clear up confusion about “what the boundary looks like” is via mathematics rather than philosophy. For example, have you understood the properties of the long line?
Great post!
Proposed solution for finite Eves (also a solution to Trumped, for finite Trumps who can’t count to surreal numbers):
After having eaten n pieces, Eve’s decision isn’t between eating n pieces and eating n+1 pieces, it’s between eating n pieces and whatever will happen if she eats the n+1st piece. If Eve knows that the future Eve will be following the strategy “always eat the next apple piece”, then it’s a bad decision to eat the n+1st piece (since it will lead to getting kicked out of paradise).
So what strategy should Eve follow? Consider the problem of programming a strategy that an Eve-bot will follow. In this case, the best strategy is the strategy that will lead to the largest amount of finite pieces being eaten. What this strategy is depends on the hardware, but if the hardware is finite, then there exists such a strategy (perhaps count the number of pieces and stop when you reach N, for the largest N you can store and compare with). Generalising to (finite) humans, the best strategy is the strategy that results in the largest amount of finite pieces eaten, among all strategies that a human can precommit to.
Of course, if we allow infinite hardware, then the problem is back again. But that’s at least not a problem that I’ll ever encounter, since I’m running on finite hardware.
We can definitely solve this problem for real agents, but the reason why I find this problem so perplexing is because of the boundary issue that it highlights. Imagine that we have an actual infinite number of people. Color all the finite placed people red and the non-finite placed people blue. Everyone one t the right of a red person should be red and everyone one to the left of blue person should be blue. So what does the boundary look like? Sure we can’t finitely transverse from the start to the infinite numbers, but that doesn’t affect the intuition that the boundary should still be there somewhere. And this makes me question whether the notion of an actual infinity is coherent (I really don’t know).
My best guess about how to clear up confusion about “what the boundary looks like” is via mathematics rather than philosophy. For example, have you understood the properties of the long line?
Thanks for that suggestion. The long line looks very interesting. Are you suggesting that the boundary doesn’t exist?
Yeah, I’d agree with the “boundary doesn’t exist” interpretation.