----- Original Message -----
From: Patrick Leahy <jpl.domain.name.hidden>
To: Alastair Malcolm <amalcolm.domain.name.hidden>
Cc: EverythingList <everything-list.domain.name.hidden>
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
Received on Thu May 26 2005 - 15:49:23 PDT