In article <4778f2c3@news.povray.org>, war### [at] tagpovrayorg says...
> Tim Attwood <tim### [at] comcastnet> wrote:
> > >  Schwarzchild radius is not dependent on density, only on the amount 
of
> > > mass.
> 
> > That's only true for black holes. For normal matter it varies by densit
y,
> 
>   "The Schwarzschild radius (sometimes historically referred to as the
> gravitational radius) is a characteristic radius associated with every
> mass. It is the radius for a given mass where, if that mass could be
> compressed to fit within that radius, no known force or degeneracy
> pressure could stop it from continuing to collapse into a
> gravitational singularity."
> 
>   "The Schwarzschild radius is proportional to the mass, with a
> proportionality constant involving the gravitational constant and the
> speed of light."
> 
> http://en.wikipedia.org/wiki/Schwarzschild_radius
> 
>   No mention of density anywhere, only of mass. The Schwarzschild radius
> of a mass can be calculated directly as the mass multiplied by a
> *constant* (which is 2*G/c^2). Volume doesn't appear anywhere in the form
ula.
> 
> > if the mass of the universe is infinite it falls inside
> > it's Schwarzschild radius, since rs = 2Gm/c^2 ... and infinity divide
d by
> > a large number is still infinity.
> 
>   Which is a rather definite proof that there's a finite amount of mass
> in the universe.
> 
> > >  I call this the shakespeare-monkey fallacy.
> 
> > It's not a fallacy, it's a paradox, there's a difference. It's not poss
ible
> > to predict random events, but it's still possible to know that they wil
l
> > occur. For example if you use a simple coin toss, and toss the coin
> > forever, the distribution will be 50/50, so you can say with 100%
> > confidence that sometime in the future there will be another "heads",
> > yet at the same time you don't know at all if the next toss will be
> > a "heads".
> 
>   It's a fallacy. An infinite amount of data, even if all values have the
> same "chance" of appearing in the data, does not automatically mean that
> every possible pattern appears.
> 
Snort. But you are proving his point. In any infinite universe in which 
"every" combination possible doesn't exist, then some combinations 
**must** be identical. Worse, if you have infinite mass, but finite 
numbers of ways it can combine, then there **must** be instances where 
some large number of duplications exist. The only question is, how 
limited is the set of possible configurations, and how infinite is the 
range of places they can take place in. Obviously, the odds of getting 
an identical sequence from a set that *only* contains 50 combinations, 
given an infinite number of tries, is much more likely than one that 
requires 500 trillion identical combinations, even with an infinite 
number "tests" (or in this case, places where such might happen). While 
its true that any *single* such sequence/set of combinations are no more 
likely than any other, it is simply not possible to have an infinite 
number of such combinations, and not have duplicates. The only 
"fallacy" is if you pointed at this planet, and said that some place 
there **must be** one just like it. We could be the one single set of 
very large number of combinations of matter that *wasn't* duplicated an 
infinite number of times, just as there could be some subset of cases 
that have "never" happened. But its just not mathematically sound to 
imply that there would *never* be such duplicated. That just isn't 
possible, unless you also have an infinite number of ways that it "can 
be" combined. Simple economies of scale imply that the smaller the 
object you are describing, the less "possible" it is for it not to exist 
some place else, in such a universe.

>   The proof is rather simple: For example, the ordered set of natural
> numbers is an infinite set where each value has an equal "chance" of
> appearing, yet eg. the pattern "2 1" never appears.
> 
Again, only if you are looking at it in terms of *each* number being its 
own unique thing. However, if you presume that the "set" of possible 
forms was 0-9, and that there is simply an infinite number of 
combinations, then obviously you *do* get "2 1", in 21, 210, 211, 212, 
etc. Your example assumes an infinite number of unique "combinations", 
in an infinite set of possible combinations. That is true. If you found 
an identical Earth, you might not find one that is both identical *and* 
in a solar system with the same number of other planets, comets, 
asteroids, constellations, etc. Something **would** be different. But 
that doesn't mean that the "object" of your examination "isn't" nearly 
identical, save for those things that are not directly "part" of it, 
which make it different, any more than you can claim that the sequence 
21 doesn't happen in 213, just because there happens to be a 3 in it.

Or, to put it in terms that one cartoon did, just because an infinite 
number of monkeys, typing randomly or otherwise, on an infinite number 
of type writers, might never reproduce Shakespeare, that doesn't mean 
they wouldn't reproduce Terry Pratchett instead. Your setting an 
artificial standard from the beginning, implying that one improbable 
solution doesn't have to happen. Its way different than saying that 
**no** improbable solution will ever happen.

-- 
void main () {

    if version = "Vista" {
      call slow_by_half();
      call DRM_everything();
    }
    call functional_code();
  }
  else
    call crash_windows();
}

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