On Mon, Apr 05, 1999 at 07:36:35PM +0000, Nick Bostrom wrote:
> What are these grounds? (You referred to "various paradoxes"--which
> ones do you have in mind?)
I can think of several paradoxes and they all relate to the fact that averages taken over the universe are not guaranteed to converge if the universe is infinite. Expectations are a kind of average, and they do not necessarily converge either. Here is an example. Suppose you are offered a bet where you can win or lose $1. You are in an infinite universe with an infinite number of people in your situation and you can't tell which of them you are. Some of them are potential winners and some of them are potential losers. The (potential) winners and losers are distributed as follows: 1 winner, followed by 2 losers, followed by 4 winners, followed by 8 losers, and so on. The paradox is that if you try to compute an expected payoff for the bet under the assumption that you are equally likely to be any of these people your computation won't converge.
The only way I can see to get around the paradox is to assume that you are a priori more likely to be near the center/beginning of the universe (or some other point), and that's what I meant by the preferred position. The exact choice of the preferred position, how "near" is defined, and how much more likely you are to be near it should all be part of the hypothesis that you are considering.
> The Big Bang is a singularity but not really a position.
> Immediately after the Big Bang, if the universe is open or flat, the
> universe was spatially infinite. So if you assign number 1 to the Big
> Bang, what spacetime point is number 2?
I wasn't being very precise when I said the conventional model has a preferred position which is the Big Bang. What I meant is that the Big Bang is a natural choice for the preferred position. There are many ways to define "near" and thus to pick point number 2. The simplest would be to to pick the point that comes immediately after the Big Bang in the rest frame of the universe.