I wanted to add a few points to my earlier posting about how to derive OM measure in a Schmidhuberian multiverse model.
The method is basically to take all the universes where the OM appears and to sum up the contribution they make to the OM measure. However, the key idea is that this contribution has two components. One is the measure of the universe. The greater the measure of the universe, the greater the contribution to OM measure. But the other is the fraction of the universe that is involved in the OM. This means that a smaller universe that contains an OM gives a greater fraction of its measure as its contribution to the OM measure. Smaller universes make more contribution than larger ones. This last step may seem ad hoc but in fact it can be seen in a very natural way. It can be thought of as a two step way to output the description of a given OM: first write a program to output a universe with the OM in it, then write a program to take that universe and output the OM. We can think of combining these two programs into one: write a program that outputs the OM. Then, the sum of the measure of all such programs is the measure of the OM. That last sentence is merely the definition of measure in a Schmidhuberian context - the measure of anything is the fraction of all programs that output that thing. It IMPLIES the formula I described for downgrading a universe's contribution to an OM by virtue of the relative size of the OM compared to the universe. It can be said that we have derived and proven that relationship by assuming this fundamental definition of measure. Note that we could also write a program to output an OM without regard to creating a universe first. However, I believe that at least for observers like us, it will always be a much simpler program to first create a universe and then find the OM in it. This lets evolution work and everything is simple. Ultimately, this allows the AUH (all universe hypothesis, ie the multiverse exists) to JUSTIFY the belief that we are not brains (or OMs) in vats, that the universe is probably real. Okay, so that's just restating what I had before in different words, explaining it from a different perspective that might be more obvious. Here are a couple of interesting additions. First, what about our universe? Why is it so damn big? If the measure of an OM is smaller in a big universe, the AUH should predict that the universe is no bigger than it needs to be. Yet, looking around, our universe looks a lot bigger than necessary. There's a lot of wasted space. I conclude that it is likely that the universe is not in fact much bigger than it needs to be. It actually needs to be as big as it is. This might imply that intelligent life is extremely rare in universes like ours. Only by creating a truly enormous universe can we have a good chance of creating observers. Let me expand on this a little. All universes exist. Some have complex laws of physics and some are simple. Some have complex initial conditions and some are simple. Physicists believe that our universe is relatively simple by both measures. The laws of physics are not completely understood but the ones we know have a very simple mathematical formulation. And the initial conditions also appear to represent a very smooth and uniform condition immediately after the Big Bang. The bottom line is that you would not have to write a very big program to simulate our universe. Yet, even with these simple laws, our universe supports life that can evolve into consciousness. That's pretty amazing, maybe. What are the odds that another universe with equally simple laws could do so? We know that our own physical laws appear to be relatively "fine tuned" such that even a tiny change in various properties would cause life as we know it to be impossible. That suggests that maybe it is not so easy to have life. Maybe almost no universes with laws as simple as ours create life. And, maybe life is not all that easy to create even in our universe. What if life, at least intelligent life, is overwhelmingly unlikely, even in a universe as well suited as our own? Maybe we need ten billion light years' worth of galaxies, stars and planets in order to have a decent chance of evolving life. Maybe, in short, our universe is as big as it needs to be, given our laws of physics, to allow life to evolve. There may be other sets of laws of physics that would be more fecund, where life could evolve more easily. Those might get by with smaller universes. But if so, the AUH would predict that such universes would have much more complicated laws of physics and/or initial conditions than our own. Otherwise we would live there. Given that the universe is as big as we see, and given the AUH, we can predict that it is not full of intelligent life. We can predict that there should be almost no other intelligent civilizations within the universe. This then solves the Fermi paradox - where are the aliens. There are no aliens, not for cosmological distances. This leads to my second point: what about infinities? In some models, our own universe is infinite in size. Tegmark's level 1 multiverse postulates a physically infinite space. His level 2 sees our universe as an infinite bubble inside a larger chaotic inflation region, with infinite numbers of other bubbles. How could my proposed formulation of OM measure work if the universe is infinite in size? How can we determine the fraction of universe measure dedicated to an OM if the OM is infinitesimal in size compared to the universe? I have two thoughts about this. One is that if the universe is truly spatially infinite, any OM should be repeated infinite times, as Tegmark predicts. Therefore the OM still occupies a finite fraction of the universe resources, and we can calculate that fraction by taking the limit as spatial size goes to infinity. I know that some people don't like this limit approach, they get upset by trying to divide one infinity by another, but in practice this method seems sensible and produces a reasonable value. The other approach focuses on a paradox between level 1 and level 2. If the level 1 universe is infinite in extent, where are the level 2 universes? Other dimensions? That doesn't really work, physically. Here is how I understand it. In physics, time and space are relative. >From the level-2 perspective, our universe is finite in size. However, it is constantly and eternally growing. From within the universe, the level-1 perspective (our point of view!) time and space have gotten shifted so that the finite-size but infinite-time outside view becomes infinite-size from the inside view. I've seen drawings of this. It is consistent to have a universe that looks spatially infinite but from the outside is finite in size but constantly growing. It's all a matter of event horizons and such. This provides a method to at least partially solve the infinite-size paradox. From the level 2 perspective, the universe is no longer infinite in size, however it is infinite in time. This means that it takes only a finite amount of computation to simulate the universe up to any given point in space and time. We can simulate the past light cone of any point in spacetime within our universe using a finite amount of computation, even though the universe looks spatially infinite from each point. This allows us to again apply our program to find OMs within the output of the program creating a universe. Each OM appears after only a finite length of output. And the farther out we go, the longer the OM-finding program is going to need to be, because it has to localize the OM within the output of the universe program. That means at some point we can ignore further OM instances appearing in the tape as having negligible measure (he says, waving his hands furiously) because the program to find them would have to be so big. In effect what we are saying is that there is a nonuniform measure over spacetime in our own universe, one that tails off to zero in space and time. Now, I'm not 100% sure how that works spatially, it seems to suggest that there is a point in space that has more measure than all others, which doesn't seem very physical. I'll have to think about it. But it is basically consistent with the general Schmidhuber principle that measure of X is 1/2^KC(X), applied to OMs. If we live in a level 2 multiverse, as our best physical theories suggest, then the measure of the OMs in that multiverse have to work in a method similar to what I have outlined here. Hal Finney
