Jesse Mazer writes:
There are two separate probabilities to consider here. One is the probability (3/4, as you show) that civilization will never break down if implemented on a computer with behaviour as specified above. The other is the probability that the actual hardware will work according to specification. I don't think you should conflate the two, effectively arguing that the hardware will work to specification because that is part of the specification!
I don't think I ever said anything about the probability involving software only. If you have a distributed computing network (such that destroying any part of it won't cause a global breakdown), and more and more of the universe is constantly being gobbled up and converted into computing power, then perhaps the probability of all the hardware in the universe breaking down would decrease geometrically as well, on average. Assume that when I talk about the probability of all copies of you being destroyed decreasing like 1/8+1/16+1/32+..., this probability takes into account all possible causes of failure, including software problems, destruction of hardware, and even stuff like the possibility that some other enemy groups of A.I.'s will attempt to erase all copies of you.
Returning to the original question, once you have settled into your new home, what is to stop all your friends disappearing, as before? The computer will try to prevent this from happening, and you could probably try the geometric series trick again (i.e. decreasing probability that your friends disappear), but in this case there will be nothing tying you to those ever-rarer branches where the hardware works as it is supposed to.
But my point is that it doesn't necessarily have to be a matter of "ever-rarer" branches--even aside from quantum immortality, it might be true that in 3/4 (or whatever) of all branches stemming from a given point in time, any A.I. around at that time will have at least some copies around in the giant computing network forever.
You seem to be treating the proposed ever-decreasing failure rate per clock cycle as if it is something that will just happen inexorably once the denizens of the far future decide to build this computer. You may as well say that in the future, there will be computers with a mean time between failure of 10^10^100 years, or whatever arbitrarily large number you choose. The problem is not in conceiving of such super-machines, it is in the details of design and implementation. I imagine that in the future there may be multiple attempts to build computers which will squeeze an infinite period of subjective time into a finite period of real time, in the way you have described, and like any other engineering project, the success rate will increase with increasing experience and resources, but even the "last gasp" effort in the moment before the big crunch will only succeed in an infinitesimally small proportion of multiverse branches.
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