Stathis Papaioannou wrote:
Now, look at p(n) again. This time, let's say it is not k, but a random real number greater than zero, smaller than 1, with k being the mean of the distribution. At first glance, it may appear that not much has changed, since the probabilities will "on average" be the same, over a long time period. However, this is not correct. In the above product, p(n) can go arbitrarily close to 1 for an arbitrarily long run of n, thus reducing the product value arbitrarily close to zero up to that point, which cannot subsequently be "made up" by a compensating fall of p(n) close to zero, since the factor 1-p(n)^(2^n) can never be greater than 1. (Sorry I haven't put this very elegantly.)
p(n) *can* go arbitrarily close to 1 for an arbitrarily long period of time, but you're not taking into the account the fact that the larger the population already is, the more arbitrarily close to 1 p(n) would have to get to wipe out the population completely--and the more arbitrarily close a value to 1 you pick, the less probable it is that p(n) will be greater than or equal to this value in a given generation. So it's still true that the probability of the population being wiped out is continually decreasing as the population gets larger, which means it's still plausible there could be a nonzero probability the population would never be wiped out--you'd have to do the math to test this (and you might get different answers depending on what probability distribution you pick for p(n)).
It also seems unrealistic to say that in a given generation, all 2^n members will have the *same* probability p(n) of being erased--if you're going to have random variations in p(n), wouldn't it make more sense for each individual to independently pick a value of p(n) from the probability distribution you're using? And if you do that, then the larger the population is, the smaller the average deviation from the expected mean value of p(n) given by that distribution.
The conclusion is therefore that if p(n) is allowed to vary randomly, Real Death becomes a certainty over time, even with continuous exponential growth forever.
I think you have any basis for being sure that "Real Death becomes a certainty over time" in the model you suggest (or the modified version I suggested above), not unless you've actually done the math, which would likely be pretty hairy.