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?
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.