Max Tegmark in his "Theory of Everything" paper gives a method for computing probabilities of statements of the form "Given that I haved perceived Y now, I will perceive X after a subject time interval t." At first I thought this method seems natural and intuitive (as long as we replace the uniform prior with the universal prior), but there seems to be some hidden paradoxes. Consider the following experiment:
At time 0 the experiment starts. At time 1 a coin is flipped and the result observed by the experimenter. At time 2 a second coin is flipped and the result observed by the experimenter. At time 3 the experimenter is duplicated if and only if both coins show heads. Now suppose the coins are fair so that the experimenter would predict at time 0 that at time 1 he will observe heads with probability 1/2. Applying Tegmark's method, he would also predict that at time 3 he will observe two heads with probability 2/5. But suppose he has already observed heads at time 1, then he would predict that with probability 2/3 he will observe another head at time 3. So he is left with the following beliefs at time 0: 1. At time 1 I will observe heads with probability 1/2. 2. If I observe heads at time 1, I will observe another head at time 3 with probability 2/3. 3. I will observe two heads at time 3 with probability 2/5. But these three statements contradict each other since 1 and 2 together implies 3a. I will observe two heads at time 3 with probability 1/3. I don't know what to make of this... anyone have any ideas?