Re: reasoning under computational limitations

Nick Bostrom (
Thu, 8 Apr 1999 20:51:01 +0000

Wei Dai wrote:

> I would say the hypothesis that I am equally likely to be any of an
> infinite number of observers is logically inconsistent. This is simply
> because there is no uniform distribution on an infinite set of discrete
> events. The universe may contain an infinite number of observers, but if
> that is true I can't be equally likely to be any of them. This is really a
> very simple and obvious idea.

The idea is simple, but it is not so obvious that it is right. I've been thinking about the possibility of using nonstandard analysis to deal with these cases. One would assign an infinitessimal prior probability to each of the hypothesis "I am the n:th human".

> > If you knew where you were, maybe you could define the preferred
> > position to be the place where you are. But in the case we are
> > considering, you don't know where you are, and any choice of a
> > preferred point seems equally arbitrary.
> I don't understand this. The preferred point is supposed to figure into
> your a priori distribution for where you are. If you already know for
> certain where you are, the preferred point is no longer relevant. The
> preferred point is like a physical constant, it is somewhat arbitrary but
> like other physical constants it has to be part of a complete theory of
> an infinite universe.

But think about what "a preferred point" means in this context. It means that you are more likely to find yourself near spatial point A than B, even though there may be somewhat more people around B. Presumably the preferred point is the same for everybody, since you say they are like physical constants. So everybody should think they are probably around B. But if everybody follows this recommendation, there will be more people who are wrong than if they just use the ordinary self-sampling assumption and think they are more likely to be around A. And all this would be known to everybody. What justtification, then, is there for following your rule?

> > > 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.
> >
> > I think there is an infinity of such points, and because of quantum
> > randomness, those points would (with prob 1) house an infinity of
> > consciousness-instances.
> I don't understand this either. How can there be a infinity of points at
> one Planck time after the Big Bang at the center of mass of the universe?

Because there is no center of gravity if the universe is spatially infinite (and roughly homogeneous).

Nick Bostrom Department of Philosophy, Logic and Scientific Method London School of Economics