On 5/20/2012 1:31 PM, Stephen P. King wrote:
My point is that for there to exist an a priori given string of numbers that is equivalent our universe there must exist a computation of the homomorphies between all possible 4-manifolds.


Why?

Markov theorem tells us that no such homomorphy exists,

No, it tells there is no algorithm for deciding such homomorphy *that works for all possible 4-manifolds*. If our universe-now has a particular topology and our universe-next has a particular topology, there in nothing in Markov's theorem that says that an algorithm can't determine that. It just says that same algorithm can't work for *every pair*.

therefore our universe cannot be considered to be the result of a computation in the Turing universal sense.

Sure it can. Even if your interpretation of Markov's theorem were correct our universe could, for example, always have the same topology, or it could evolve only through topologies that were computable from one another? Where does it say our universe must have all possible topologies?

Brent

It is well known that the act of defining an exact "time slice" is a computationally intractable problem, the Cauchy surface problem <http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&sqi=2&ved=0CE8QFjAB&url=http%3A%2F%2Fwww.tapir.caltech.edu%2F%7Elindblom%2FPublications%2F10_CommMathPhys.61.87.pdf&ei=G1O5T9y1NYG69QT036GoCg&usg=AFQjCNGQxEJa9DEFbyKnXiub-nS7zvPksw&sig2=9za6duwSbfQp-yZ_Cj6vzA>. Physicists use approximations and cheats to get around this intractability.

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