On Wed, Jan 21, 1998 at 10:33:27AM -0800, Hal Finney wrote: > I'm not sure I follow you here. Are you suggesting that our observations > of the universe's history might be in error, that we might be instantiated > for a single instant with all of our memories of the past being illusions? > We could write down the state of the universe at this instant as a long > string using a simple mapping, and at some point the counting TM will > emit this string. At that instant we will all exist with our memories > of the past, but none of that past will actually have happened. Is this > right?

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Yes, that's what I'm saying. > The problem then is not so much with the objective question of whether we > live in the counting universe, which an outside observer could easily > answer, but rather with the difficulty of us as residents in the universe > knowing the answer. > > A more extreme case occurs if the counting program works in trinary, > using the 0/1/comma alphabet. Now it outputs not only all numbers, it > outputs all possible comma-separated sequences of numbers. It eventually > outputs not only every possible state, but every possible sequence of > states, including the entire history of our universe. This history would > be a subset of an enormously larger output string, all created with a very > short (and therefore a priori probable) input program. > > All I can suggest about these problems is that they would imply that the > universe will shortly cease to behave lawfully. When we don't observe > this, we can reject this possibility. Otherwise we are forced to say > that all our perceptions are illusions, that our memories are false, > that we ourselves may be simply memories of some future self. Logically > we can't rule this out, but it does not seem reasonable. If we lived in > the counting universe, there is no reason why the universe should seem > lawful in the sense we see. Logically we can't rule this out, but we should be able to rule it out probabilisticly. My point is that I don't see how to do it under Schmidhuber's interpretation. > You solution is somewhat different in nature from Schmidhuber's. > Your set of possible universes is more structured than his. You have > a coordinate system, you have regions (which may imply a certain amount > of continuity in the coordinates). He had binary strings to represent > the universe state, which would allow for a wider set of possible > universes. The coordinate systems and regions are just interpretations to help our intuition. The procedure that I gave for computing probabilities does not depend on any structure in the coordinate system. If its not easy to interpret the input to a TM as coordinates in a universe, then it would be hard for us to think about that TM as the sort of universe we're familiar with, but it doesn't matter in the probability computations. > I'm not sure you can fully avoid the mapping problem with your approach. > There is still a mapping involved in the choice of coordinates. > With a sufficiently exotic coordinate system, I suspect we could map > our universe's state to the output of a trivial program. Remember that the prior probability of a region is related to the program length plus the coordinate length. If you have a universe with an exotic coordinate system, the lengths of the coordinates would be long and so the regions in that universe would have small priors.