>-----Original Message----- >From: Patrick Leahy [mailto:[EMAIL PROTECTED] >Sent: Thursday, May 26, 2005 10:21 AM >To: Alastair Malcolm >Cc: EverythingList >Subject: Re: White Rabbit vs. Tegmark > > > >On Thu, 26 May 2005, Alastair Malcolm wrote: > >> An example occurs which might be of help. Let us say that the physics of >> the universe is such that in the Milky Way galaxy, carbon-based SAS's >> outnumber silicon-based SAS's by a trillion to one. Wouldn't we say that >> the inhabitants of that galaxy are more likely to find themselves as >> carbon-based? Now extrapolate this to any large, finite number of >> galaxies. The same likelihood will pertain. Now surely all the >> statistics don't just go out of the window if the universe happens to be >> infinite rather than large and finite? >> >> Alastair > >Well, it just does, for countable sets. This is what Cantor showed, and >Lewis explains in his book. Cantor defines "same size" as a 1-to-1 >pairing. Hence as there are infinite primes and infinite non-primes there >are the same number (cardinality) of them: > >(1,3), (2,4), (3,6), (5,8), (7,9), (11,12), (13,14), (17,15), (19,16) etc >and so ad infinitum > >You might say there are obviously "more" non-primes. This means that if >you list the numbers in numerical sequence, you get fewer primes than >non-primes in any finite sequence except a few very short ones. But in >another sequence the answer is different: > >(1,2,4) (3,5,6) (7,11,8) (13,17,9) etc ad infinitum. > >In this infinite sequence, each triple has two primes and only one >non-prime. Hence there seem to be more primes than non-primes! > >For the continuum you can restore order by specifying a measure which just >*defines* what fraction of real numbers between 0 & 1 you consider to lie >in any interval. For instance the obvious uniform measure is that there >are the same number between 0.1 and 0.2 as between 0.8 and 0.9 etc. >Why pick any other measure? Well, suppose y = x^2. Then y is also between >0 and 1. But if we pick a uniform measure for x, the measure on y is >non-uniform (y is more likely to be less than 0.5). If you pick a uniform >measure on y, then x = sqrt(y) also has a non-uniform measure (more likely >to be > 0.5). > >A measure like this works for the continuum but not for the naturals >because you can map the continuum onto a finite segment of the real line. >In m6511 Russell Standish describes how a measure can be applied to the >naturals which can't be converted into a probability. I must say, I'm not >completely sure what that would be good for. > >Paddy Leahy
I agree with all you say. But note that the case of finite sets is not really any different. You still have to define a measure. It may seem that there is one, compelling, natural measure - but that's just Laplace's principle of indifference applied to integers. The is no more justification for it in finite sets than infinite ones. That there are fewer primes than non-primes in set of natural numbers less than 100 doesn't make the probability of a prime smaller *unless* you assign the same measure to each number. Brent Meeker
