From: eugene@wilbur.nas.nasa.gov (Eugene N. Miya)
Newsgroups: rec.backcountry
Subject: Re: Stars (An) Insight
Date: 1 Oct 92 00:38:07 GMT
Organization: NAS, NASA Ames Research Center, Moffett Field, CA

In article <1992Sep30.221219.3745@iscnvx.lmsc.lockheed.com>
tilley@sunfse.ese.lmsc.lockheed.com writes:
wayne>Lying staring at the stars last weekend I had an insight.
wayne>It was one of those clear, moonless, wilderness nights and the
wayne>stars were uncountably many.

enm>Not to burst your romantic bubble, there are approx. 4K stars visible to
enm>the human eye (both hemispheres, not all at one time).  Even the best

w>Oh Eugene.  You're so left-brained!  (Don't feel bad.  That's a line
w>an old room mate used to apply to me all the time.)

tom>To be "left brained" for a moment, the quoted passage does not claim
tom>that the visible stars were uncountably many: only that "the stars"
tom>were. :-)

Let's see, we can analyse this for the astronomy: Olber's Paradox
(If there are so many stars, why is the sky full of dark areas?) or
we can take the mathematical analysis/physics approach:
Countability is defined as the mapping of set elements to the natural numbers
(1,2,3,4...), an enumeration (not the programming language type).

I always did like astronomy, but I should try the math first.

Each star represents a discrete element, therefore we can enumerate, therefore,
the set is countable.  Since the universe is regarded as finite, one can see
that we have a finite number of discrete elements, therefore the set of all
stars is countably finite.  The part about discrete stars can be proven
by standard topological mathematical analysis:

Take a point $P sub 0$ (I choose eqn(1) as the reader may not have
familiarity with TeX) as the center of the star.  For a point P near
$P sub 0$, there exists a neighborhood (designated $N sub epsilon (P)$)
for distance $epsilon > 0$ for some Euclidean measure such that
the set of points P are 1) outside the star [call this "space"],
2) inside the star (interior), or 3) are the stellar boundary.

1) The $N sub epilson (P)$ where NO PART of the neighborhood
is part of the star as the distance $epsilon$ approaches 0
($lim as epilson -> 0$),
2) EVERY element of $N sub epsilon (P)$ has star stuff
inside it as the distance $epsilon$ approaches 0, or
3) every $N sub epilson (P)$ neighborhood has BOTH space and star stuff inside
it [this is the boundary] inside it as the distance $epsilon$ approaches 0.
Since the Universe has known finite volume, any subset part must also have
finite value.  Therefore the discrete element parts as defined above
must be countable.

Now, we have to deal with the observable, hence visible.
Corollary, this must hold for dark matter.  Dark matter must exist in
the known universe.

Proof of Olber is left as an exercise for the reader.

And furthermore.....

>This exchange reminds me of Ray Bradbury's introduction to his "Complete
>Book of Ray Bradbury Poetry", or something to that effect (pub. early
>1980's).  He related that he had a line about how "the birds have not
>seen the stars", and was informed by Carl Sagan (who was on a boat with
>him at the time) that there are birds that migrate by the stars.  He
>came up with "maybe some bird has felt Orion's rise, but see .. but really
>see and know?".

I hope you have seen or will see 
	    s
	A Co^mic View by Pacific Data Images.


	;^)

--eugene miya, NASA Ames Research Center, eugene@orville.nas.nasa.gov
  Acting Associate Editor, Software and Publication Reviews
  Scientific Programming
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