Devin Harris wrote: > Again, the question of how infinite is the Universe that > contains MWs. Or said otherwise, how vast or how ruled is > the possible world, assuming all possibilities exist?
I posted about this before, in http://www.escribe.com/science/theory/index.html?mID=674. I think that what Devin describes in his last post is correct: that the cardinality of the structure in which we find ourselves must be, in some sense, be infinitely infinite. It must be Aleph-infinity, if this makes any sense. A trivial application of the SSA indicates this. I've thought about this some, but I am not a mathemetician, so I'd appreciate any feedback. For any arbitrary (finite or infinite) set S, the Power Set of S is denoted *S, and is the set of all subsets of S. For example, if S contains {1, 2, 3}, then *S would be { {}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} }. *S is called the "Power Set" because, for finite sets: |*S| = 2 ^ |S|, where |S| is the cardinality of S. For infinite sets, I believe that this power equation is defined in terms of |*S|. Cantor's theorem states that, for any set (finite or infinite), |*S| > |S|. This provides a mechanism for generating infinite sets with ever increasing cardinality. defined in terms of |*S|. Now, the cardinality of the power set of integers is c, the cardinality of the real numbers. I think of this intuitively by imagining that every real number can be expressed as an infinite number of binary digits, for example, ...1101001000101110110101.11101010001010101001... Each digit can have one of two values, and there are |Z| digits, so c = 2^|Z|. Does anyone know what the cardinality of the branches in the traditional MWI is? By traditional, I mean that from the Schrodinger's Equation. It seems plausible to me that it's 2^c, by an intuitive reasoning similar to the above. I think it depends on whether the universe is finite or infinite in the number of particles it holds. Now, if the original line of reasoning is correct, then I predict that further breakthroughs in physics will exhibit, by some mechanism (this is obviously highly speculative) that the cardinality of the "branches" is of a new order. By the SSA, we must find ourselves in universes with physical laws that admit the greatest number of SASs. Perhaps this feature has already been exhibited by some of the more advanced quantum theories out there - I admit ignorance of QFT in general. The only mechanism that I can imagine is some sort of infinite regress with regards to the structure of the universe. That is, I would guess that no matter how far we dig into the underlying structure of matter, we will always uncover deeper layers. -- Chris Maloney http://www.chrismaloney.com "Knowledge is good" -- Emil Faber