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Re: Problems with Infinity?
Posted:
Feb 26, 2013 4:44 PM
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:: 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" <b.scott@csuohio.edu>
: 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
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