Re: Attempts to Refute Cantor's Uncountability Proof?

matt271829-news@xxxxxxxxxxx wrote:


Hatto von Aquitanien wrote:

As a side note. I had a friend raise a very simple objection to the
notion
that every finite region of R^2 has the same number of points. His
argument was that if we draw a circle and confine a bunch of points in
it,
then draw a larger circle around it. there are points in the larger
circle
which are not in the smaller one. It would seem to follow that there are
more points in the larger circle than in the smaller one. I have never
fully convinced myself that his reasoning is lacks merit. Yes, I know
I'm not supposed to entertain such doubts, but....

Take a *** of rubber, stretch it, and draw a circle on it (that's
your larger circle). Then let the rubber contract so that the larger
circle becomes the smaller circle. Are there any points in the larger
circle that don't become points in the smaller circle?

Take a *** of rubber and paint a circle on it, and then stretch it.
Either the paint will fragment because it doesn't stretch with the rubber,
or it will become thinner and its color will therefore be less intense. I
fully understand the reasoning that says there are as many points in the
smaller circle as there are inside the larger circle, but that doesn't
quell some intuitive reluctance I have to completely dismiss my friends
counter argument.
--
Nil conscire sibi
.

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