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Jun 23rd, 2017
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  1. B1 = "We live in an LPU-Island larger than 10^10", then P(B1 | T) = 1 and P(B1 | NSU) << 1. So, P(B1 | T) >> P(B1 | NSU). That, I'll happily grant
  2. B2 = "We live in an LPU-Island no larger than 10^500."
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  5. One response to this line of thinking is to recognize that the probabilities involved in the B2 calculations violate countable additivity, specifically the value of P(B2 | T & B1), which equals zero if countable additivity does not hold, and is undefined if it does. If countable additivity is taken to hold, then this response fails. But, this has implications for the other leg of the argument, specifically the value of P(LPU | NSU & k').
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  7. If countable additivity holds, then P(LPU | NSU & k') is undefined. This can be seen by seeing the entire, infinite modal space as divided up into regions equal in size to the life-permitting region we are actually in. Given that NSU provides no reason to think that the actual universe being in any of these regions is more likely than landing in any other, the indifference principle tells us that we should assign equal probabilities to them all. Of course, there is no number we can assign to this probability such that the sum of all of them is 1. Thus, given countable additivity, the probability of the actual universe being found in any of these regions is undefined. So, the crucial premise "P(LPU | NSU & k') << 1" is false. Shoring up the LPU-Island leg of the argument by embracing countable additivity seems to come at the cost of guaranteeing the whole argument not be sound.
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