From: Helger Lipmaa <helger@saturn.tcs.hut.fi> Subject: [math-fun] Parallel Universes
http://www.sciam.com/article.cfm?articleID=000F1EDD-B48A-1E90-8EA5809EC58800...
"In infinite space, even the most unlikely events must take place somewhere."
In serious probability theory this question is typically not addressed, although I think it should be. The general situation is where there is an event whose probability is 0, but which is not impossible -- like picking a number at random from the unit interval and getting 1/3. When the experiment of picking the random number is repeated enough times, we, or at least I, would expect the probability to be 1 that 1/3 has come up at least once. One thing is for sure -- if the experiment is repeated a countably infinite (aleph_0) number of times, the the probability is 1 that every number in the unit interval will have arbitrarily nearby numbers having been picked. When attempting to apply this to the real world, one could fairly say that in an infinite universe (assuming appropriate randomness and a continuous state space), then with probability 1, everything that might happen in a given finite portion of spacetime is approximated with arbitrary precision -- .somewhere. (This ignores the possibility of black holes, alternative universes, and just about all the weirdness that is likely to be true.) Back to the unit interval [0,1). Suppose we were to pick a number at random from [0,1) an infinite number of times -- let's call the infinite cardinal "beta" (which of course satisfies beta >= aleph_0). Then what is the chance that we never picked 1/3 ? (Where 1/3 could clearly be replaced by any other x in [0,1).) Heuristically, this probability should be (1 - 1/c)^beta, where c = continuum = # of points in [0,1). This expression is meaningless in ordinary cardinal arithmetic but can be made to make sense in, say, the surreal (Conway) numbers. In surreal numbers, the nearest real number to (1 - 1/c)^beta is 0 if beta < c, 1 if beta > c. I'm not sure whether (1 - 1/c)^c = 1/e (to the nearest real), but wouldn't that be nice? Let's call this probability P(beta). However, picking a random number from [0,1) once for each t in [0,1) -- which heuristically yields 1/e for P(c) -- leads to a paradox. Since the index set {t} = [0,1) is bijectively equivalent to [0,2), this means that doing the experiment once for each t in [0,1) should yield the identical probability as doing it once for each t in [0,2). This means that P(c)^2 = P(c), so P(c) = 0 or 1, and NOT 1/e. This is frustrating to me, since morally it "should" be true that P(c) = 1/e. --Dan