Robin Hanson wrote:
> Nick B. writes:
> >The conclusion of the self-indication axiom is even stronger:
> >if infinite universes are possible, then it is a priori *certain*
> >that the universe is infinite.
>
> Er, don't you have to assume a non-zero probability for infinite
> universes, and then don't you only get a probability one for infinite?
> Things can be possible and have zero prob, and can be prob one without
> being certain.
Can they? I used to say that too, but I reflected a little bit on it recently and it's not so clear that that is the right thing to say. At any rate, read "prob one" and "prob zero" if you want -- you still get a heavy conclusion. Do you accept it? That sitting in your armchair you can ascertain that the probability that the universe is infinite is one? (And it's hard to see how this conclusion could be changed by any empirical evidence.)
> let me explain what I mean by formalize.
Ok, I'll make a stab at it. (It's somewhat late here but curiosity drives me to stay up and watch Clinton address the nation.)
Within this framework I think one could state the DA as follows:
Suppose there are two possible worlds, W1 and W2. In W1 there are two consequtive states, S1 and S2; in W2 there is only one state, S1, whereafter the world ends.
There is one agent, a, in each of these possible worlds. a can be in state a(S1) or in a(S2).
The "possible states" for the a's states are: P(S1)={(S1, W1), (S1, W2)} and P(S2)={(S2, W1)}.
We can assume that the prior probabilities are equal for W1 and W2.
Suppose RH-now is a being in state S1. In S1 you have information I. Question: What is Prob(W1|I)?
Some would say 1/2. But the doomsdayer claim that that is to overlook the fact that you should conside RHnow as a random sample from all actual observer-moments. The doomsdayer defends this claim like this:
Consider a hypothetical agent, ah, which is like a but has a reduced
information set I'. Suppose that relative to I', the possible states
for a are {S1, S2}, and that RH-now
thinks that the rational probabilities for ah are
Probah(W1|I')=Probah(W2|I')=1/2.
But consider the conditional probability Probah(I' & "a is in state S1"|W1). Suppose RH-now thinks that the rational estimate of this is 1/2. Then, by Bayes theorem, RH-now should think:
Probah(W1|I'& "a is in state S1") <1/2
But then, in the rational opinion of RH-now, Probah(W1|I)<1/2, since I = I'&"a is in state S1". Thus, if RH-now prescribes the estimate Prob(W1)<1/2 for somebody just like him and with exactly the same information aas himself, surely RH-now should himself believe that Prob(W1)<1/2!