Infinity

Eric Wald eswald at brainshell.org
Fri Jun 3 23:12:51 MDT 2011


On Sat, May 28, Levi Pearson wrote:
>
> Now, if you want to know a set with a higher cardinality than R, I
> can't help you there.

The set of all squiggles on the plane; or, equivalently, the set of all
sets of rational numbers.  That seems to be the cardinality of the set
of all n-dimensional objects, for any finite n (neglecting the physical
restraints of elementary particles), and probably the set of points in
an infinite-dimensional space.

I'm fairly certain that the set of all infinite-dimensional objects is
larger still.  However, I can't say for sure which cardinality holds the
set of all finite-dimensional objects.

Blame "Here's Looking at Euclid" for at least part of that line of
thought.  My wonderful wife picked up a copy for me; it's a very
well-written book, interweaving descriptions of several interesting
mathematical concepts with the history of various mathematicians who
studied them.  It makes the point, for example, that the general decline
in religion's power over the last few centuries may very well be due to
gambling...

- Eric


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