Imagine that Fred has all ten toes at 10 am, and there are infinitely many grim reapers. When a grim reaper wakes up, it looks at Fred, and if he has all his ten toes, it cuts one off and destroys it; otherwise, it does nothing. There are no other toe-cutters around.
Suppose, further, that grim reaper wake-up times can be set by you to any times between 10 and noon, endpoints not included. If you set the activation times to be such that there is a first activation time after 10 am (e.g., the nth reaper wakes up 60/n minutes before noon), there is no paradox of any sort. But if you set the times such that they are all after 10 am, but before every activation time there is another activation time, then… well, then logic guarantees that Fred will get a toe cut off infinitely many times and will regrow a toe infinitely many times! For without toe-regrowing, we get a paradox.
This is, of course, logically and metaphysically possible. Toes can regrow, and it is metaphysically though perhaps not physically possible for them to do so quickly. But what is amazing is that just by setting wake-up times for grim toe-cutters, we can make this miracle happen.
2 comments:
I feel that the Unsatisfiable Pair Diagnosis confuses an epistemic explanation for a more metaphysical one (not sure what term to use here). For example, if you have someone who committed a crime, the DNA on the crime scene would confirm that he did it. You could say that he committed the crime because we have found the DNA evidence. That's the explanation. However, you could also say that the explanation for him committing the crime is the internal mental reasons that led him to do it. The fact that it is a contradiction explains our knowledge of its impossibility, but there must be another type of explanation for it as well.
In your book, you object to the no room response for a number of reasons. I accept (2), (4), and probably (3) that you lay out, but I will talk about the others. One could say that an infinite number of things cannot exist in a finite space. This is because as you grow the number of things that exist in that finite space, space warps. Eventually, a black hole is created, preventing the process. So, you can't have an infinite number of grim reapers at ...1/4, 1/2, 1, because that would have an infinite number of things in a finite space. You can also deny (7) by just saying that it's impossible for an infinite number of things to be in the same spaciotemporal location. Again, it is not ad hoc because of the black hole. For (5), one could say that there can be an infinite number of things in infinite space, but they can't all causally interact with one another because of the finite speed of light. So, the reapers being ...-3, -2, -1 in space and checking an infinite number to the left is impossible. For (6), I'm not sure what to say. For example, an angel must only be able to interact with a finite region in space so that an infinite number of spacetime points don't cause the angel to do something, violating causal finitism. Could you have an immaterial mind receive information at one point and then cause something in a distant part of space, violating the speed of light? I don't think so. I'm not sure how a non-spatial contingent being would relate to time given that time and space are so interconnected. So, I think the no room explanation of causal finitism might still make sense in the end.
Logic shows that some seemingly consistent sets of conditions really are consistent and that others aren’t. It doesn’t make anything happen.
Logic shows that the usual explicit GR conditions (rephrased for toe cutting), the implicit condition that Fred’s toes don’t regrow, and a beginningless sequence of wake-up times are inconsistent.
The conditions can be modified to allow regrowth. (To avoid complications, require that regrowth is instantaneous and never happens at a GR activation time.) Logic then shows that there are outcomes consistent with all the new conditions, even for beginningless sequences of GR wake-up times. (E.g. regrow toes between consecutive wake-up times.) But it doesn’t make such outcomes happen – it just shows they are possible.
If Fred doesn’t know the wake-up times in advance, but regrows toes only in response to losing them, the paradox is not avoided. In the interval between the first time Fred notices a toe missing and regrowing it, there will have been a beginningless sequence of wake-up times, with the usual paradox.
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