> From Russell Standish Thu Feb 8 23:52:51 2001 > Guys, > I'm getting great enjoyment out of the titanic battle between > Juergen and Bruno over the meaning of the UD. I'm learning a lot from
Battle? The case is clear. You cannot battle over whether 2+2 equals 4 or 5. > the exchange, however, I must admit I do see Bruno's point of > view. His UD does seem to generate the reals (or equivalently the set > of all infinite binary strings) in countable time. However, I know Even Bruno admits this is not true. Thus his NONalgorithmic "arithmetic realism." > that infinity (like probability) is a nasty concept, that can easily > trip you up. It's easy: just don't confuse the countable set of all finite beginnings of the reals with the uncountable set of all reals, which does not exist from an algorithmic or constructive point of view. > There other ways of approaching this - for instance a finite set of > axioms, when enumerated into theorems will tell us all that can be > known about the real numbers. I cut and paste from the thread "Algorithmic TOEs vs Nonalgorithmic TOEs": Loewenheim-Skolem implies that any first order theory with an uncountable model such as the real numbers also has a countable model. None of the countably many theorems concerning the real numbers depends on the "continuum", whatever that may be. Our vague ideas of the continuum are just that: vague ideas without formal anchor. > I have sympathy for one point of Juergen's though - in the space of > descriptions (which we should agree by extension of logical positivism > is all that can be discussed), computable descriptions must have > higher measure than noncomputable ones. However, it seems to me that a > random oracle is an essential component of consciousness and free will > - why this is so I can only guess - and so the anthropic principle > constrains the interesting universe to having these. It could be that > this random oracle is simply a consequence of 1st person indeterminism > that arises through the duplicability assumption, as Bruno points out, > but then why should duplicability be necessary? There is no evidence whatsoever that we need a random oracle.