----- Original Message ----- From: Patrick Leahy <[EMAIL PROTECTED]> To: Alastair Malcolm <[EMAIL PROTECTED]> Cc: EverythingList <firstname.lastname@example.org> Sent: 26 May 2005 11:20 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!
I've got no problem with this, as far as it goes. The point I was trying to make - talking in terms of prime numbers if you prefer - is that in the circumstance I am referring to we equivalently *are* in the situation of a particular pre-defined sequence - and so relative frequency becomes relevant. Alastair