Ben Jones: Dead Christian Transhumanists' souls await the Final Day, just like all the rotting Christian corpses' souls. If you thaw out their heads and turn them back on, the souls are right where they were before, to the extent they ever were anywhere, and might as well pick up where they left off. When the head finally gets used up, the soul may be presumed to be where the other corpses' souls are.
The real question is what happens when you scan the frozen head into a non-frozen-head simulator, and start it up: can the simulation exchange packets with the waiting soul? If you start up two (or a million) simulators, can they all use it, or only the first? Or do that have to take turns, as with a time-share condo?
I think we would have to conclude that the soul (its "stamp", if you like) is encoded in details of the frozen head, and, if the non-frozen-head simulator were precise enough, you'd be running a simulation of the soul in each run, while the real soul sits twiddling its metaphorical thumbs waiting for the End Times. If the non-frozen-head simulator can simulate a non-frozen-head well enough, surely its simulation of the soul ought to be good enough too. After all, what does a soul have to do? Some people would say a cantaloupe provides a perfectly adequate simulation today, but they're probably not Christians.
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For an unbiased, science-based model of the Singularity, there is no better source than Jared Diamond, "Collapse" and "The Third Chimpanzee". The Americas and many Pacific islands have faced mini-singularies in the recent past. Among the more poignant is Easter Island, where every last tree was cut down.
Diamond identifies a round dozen worldwide crises, any one of which may cause a corresponding worldwide collapse in the next century. Among them are nuclear-weapon proliferation, water shortages, devastation of fisheries, global warming, elimination of tropical forests, and peak oil. He addresses the prospects of "technically innovating" our way out of all twelve, and the outlook is bleak.
Could a fancy AI help? Only if we followed its advice. We are already getting much better advice than we seem able to follow.