:: Similarly, the cardinality of the set of paths in n>=2 d space :: (if I'm remembering a close enough description) is bigger still.
: "Brian M. Scott" <firstname.lastname@example.org> : No, it's also 2^\omega, since by definition a path is continuous.
Do you mean points on the... ah, no wait. I see (or google) that I most likely original got that meme from Gamow's "One Two Three... Infinity". Here, some of what he says is corrected:
http://www.ii.com/math/ch/confusion/ An example of confusion is in George Gamow's One Two Three...Infinity. On page 34 he says: The sequence of numbers (including the infinite ones!) now runs: 1 2 3 4 5 ... aleph1 aleph2 aleph3 ... and we say "there are aleph1 points on a line" or "there are aleph2 different curves" ... First note that Gamow's sequence of finite and transfinite numbers is not correct. It should be: 1 2 3 4 5 ... aleph0 aleph1 aleph2 aleph3 ... Also note that he is assuming CH when he says "there are aleph1 points on a line" and GCH when he says "there are aleph2 different curves."
However, these confusions and others don't seem to invalidate the notion I had about "bigger still", since per above, the GCH seems to imply it. Or am I still missing something? Probably conflating omega with something.
Granted, even if I'm not missing and didn't conflate, it would imply I inadvertently drug the GCH in without notice, so
"Insofar as I may be heard by anything, which may or may not care what I say, I ask, if it matters, that I be forgiven for anything I may have done or failed to do which requires forgiveness." --- paraphrased excerpt from the Possibly Proper Death Litany, from Creatures of Light and Darkness
"Lord, I apologize fer that." --- Larry the Cable Guy's condensed version