A being you know for sure to be completely reliable on what it says tells you that that the universe will contain an infinite sequence of people who can be totally ordered by time of conception. The being also says that the people can also be totally ordered by their distance from the universe's center (center of mass? or maybe the universe has some symmetries that define a center) at the time of their conception. Finally, the being tells you:
- If you order the people by time of conception, the sequence looks like this: 99 people who will die of cancer, then one person who won't, then 99 people who will die of cancer, then one person who won't, and so on.
- If you order the people by distance of conception from the center of the universe, the sequence looks like this: 99 people who won't die of cancer, then one person who will, then 99 people who won't, then one who will, and so on.
Question: What probability should you assign to the hypothesis that you will die of cancer?
If you just had (1), you'd probably say: 99%. If you just had (2), you'd probably say 1%. So, do we just average these and say 50%?
Now imagine you just have (1), and no information about how things look when ordered by distance of conception from the center of the universe. Then you know that there are infinitely many ways of imposing an ordering on the people in the universe. Further, you know that among these infinitely many ways of imposing an ordering on the people in the universe, there are just as many where the sequence looks like the one in (2) as there are ones where the sequence looks like in (1). Why should the ordering by time of conception take priority over all of these other orderings?
An obvious answer is that the ordering in (1) is more natural, less gerrymandered, than most of the infinitely many orderings you can impose on the set of all people. But I wonder why naturalness matters for probabilities. Suppose there are presently infinitely many people in the universe and when you order them by present distance from the center of the universe, you get the sequence in (1). That seems a fairly natural ordering, though maybe a bit less so than the pure time-of-conception ordering. But now imagine a different world where the very same people, with the very same cancer outcomes, are differently arranged in respect of distance from the center of the universe, so you get the sequence in (2). Why should the probabilities of death by cancer be different between these two worlds?
So what to do? Well, the options seem to me to be:
- Dig heels in and insist that the natural orderings count for more. And where results with several natural orderings conflict, you do a weighted average, weighted by naturalness. And ignore worries about worlds where things are rearranged.
- Deny that there could be infinitely many people in the world, even successively, perhaps by denying the possibility of an infinite past, a simultaneous infinity and the reality of the future.
- Deny that probabilities can be assigned in cases where the relevant sample-space is countably infinite and there are infinitely many cases in each class.