Here is what I considered. Imagine for the sake of example that we are
in an infinite, non-closed universe. Shooting a laserbeam into space
would go out forever and would be considered to have an infinite length
at the end of an infinite time.
Now imagine a beam projection that is one inch wide, one inch tall, and
of infinite length. The volume of such a beam would be 1" x 1" x
infinity". It has one infinity of cubic inches within its volume.
Now imagine a beam projection that is two inches wide, one inch tall,
and of infinite length. The volume of such a beam would be 2" x 1" x
infinity". It has two infinity of cubic inches within its volume.
It seems obvious to me from a physical standpoint, that the 2 x infinity
inches volume is twice as large as the 1 x infinity inches volume. It
would take two of the former added together to equal the latter. If you
superimposed the 1 x infinity inches volume inside the 2 x infinity
inches volume, you would have 1 x infinity inches volume left over.
Here is another thought. The distance from our viewpoint to the
infinite edge of the universe is 1 x infinity. The distance in the
opposite direction to the infinite edge of the universe is also 1 x
infinity. Therefore, the widths of the unbounded universe would be 2 x
infinity. We also can measure it in three dimensions, so the volume of
the unbounded universe would be (2 x infinity) cubed.
And here is a third thought. We can conceptualize an infinite length,
but in reality we can only approach it at the speed of light. Any
infinite length, therefore, might be considered to be a length
approaching infinity at the rate of the speed of light. Maybe different
infinities can also be measured or contemplated in terms of their rate
of approaching infinity. The theoretical maximum speed of light would
be the maximum approach toward infinity. Something going half the speed
of light would be approaching infinity also, but at half the rate. This
seems to consider time as the forth dimension, and describes a function
chart approaching infinity at a different slope on the graph, or a
different rate.
I have never met anyone who even considered these ideas. My friends
always immediately rejected the ideas, claiming that all infinities are
infinite, and are all equal "by definition". It sounds like other
people on this list are considering different orders of infinity.
-- Harvey Newstrom (harv@gate.net)