Re: The logic of atheism

prabbit1@xxxxxxxxxxxxxxx wrote:
Ron Peterson <ron@xxxxxxxxxxxxxx> wrote:

prabbit1@xxxxxxxxxxxxxxx wrote:

Just happened to think of something that shows how infinities don't "follow
the rules." Any infinity of order 1 (forget the proper term) is "equal in
size" to any other infinity of order 1 (so that they can be matched in a
1-to-1 manner.) I.e. the set of all whole numbers is "as large as" the set
of even whole numbers. So a square with infinitely long sides would have
infinitely long diagonals (here, diagonal will refer to any line that
crosses the side and the middle, even if it's not corner-to-corner. Don't
know if there's a more correct term than "diagonal" but it'll work for us.)
So all the diagonals would be of equal length in the segment of them that's
wholy contained in the square and and half of each diagonal woult be equal
to half of every other diagonal (since half of infinity is infinity), thus
it would satisfy the "every point on the figure [edge] is equidistant from a
central point" part of being a circle at the same time it's a square (or ANY
infinitely large 2-D, singlying connected shape, for that matter.)

Strange but apparently true (or can someone find a flaw that I overlooked
somewhere?)

You're trying to define a square which in the infinite case would
coincide with a circle. I don't know if you can do that since a square
needs four distinct vertices and lines joining them.

A better approach would be to define a different metric on the plane so
that the coordinates of the lines describing a square would be
equidistant from the center of the square.

Hmmm...trying to visualize something like a saddle-shaped surface that had
angles instead of curved humps. Think of something like a piece of paper
folded from corner to corner and then across the other corners. Then hold it
with the middle point and creases pointing up and pushed down between the
creases. Now of course that'd still be a euclidean plane but then stretch it
so that it becomes a 4-way saddle-shaped 2D surface but still having the 4
creases joining in the center with the areas between them curving downwards.
Seems like we could wind up with the right shape there somehow? Nah, scratch
that one, just realized that a square on a saddle starts looking like a
euclidean square with the sides pushed inwards.

But actually a square can be defined as "4 equal-length line segments joined
at the ends where all 4 angles are the same and all 4 lines are the
shortest-distance path between the pairs of vertices." Well, on a sphere,
the shortest-distance path is part of a "great circle" (i.e. a
circumfrential circle.) So if you took a great circle and divided it into 4
equal segments then the segments would join at the vertices (the point at
which the two sides of an angle, in this case, a 180 degree angle, join) and
would all be shortest-path segments and the angles would all be equal in
size (all 180 degrees.) Hey, no-one said this had to be euclidian squares
and circles :)

Scratch the above. Actually the OP DID say "euclidean space." <sigh>

Oh, well, back to the drawing board<laugh>

--
Mike

-------------------------------
"Our enemies are innovative and resourceful, and so are we. They never stop
thinking about new ways to harm our country and our people, and neither do
we," George W. "Shrub" Bush Aug 5, 2004
.

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  • Re: The logic of atheism
    ... Any infinity of order 1 (forget the proper term) is "equal in ... So a square with infinitely long sides would have ... So all the diagonals would be of equal length in the segment of them that's ...
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