I think Gamow's intention on was to talk about arbitrary, rather than continuous, 'curves' which indeed have cardinality beth_2. On 12/7/12, Adam P. Goucher <apgoucher@gmx.com> wrote:
Oh dear! The first box is definitely correct, the third box is incorrect, and the second box depends on the continuum hypothesis (as does the caption).
Aleph_1 is identified as "the number of all geometrical points on a line", and the caption called Aleph_0, Aleph_1 and Aleph_2 "the first three infinite numbers" (which implies there are no other infinite numbers in between).
Aleph_1 is defined to be the second infinite number, which is only the same as Beth_1 (cardinality of the reals) if you assume CH.
Also, the number of curves (by which I mean countable unions of continuous curves) is just Beth_1, same as the cardinality of the reals. This includes everything you can justifyably call a curve (including the boundary of the Mandelbrot set!).
Beth_2 (= 2^Beth_1 = 2^2^Aleph_0) is the number of ways you can colour the plane with two colours (which I think is what Gamow meant by 'geometrical curves').
Yes, this is one of those things which are obviously false but actually true. Then there are things which are obviously either true or false, but turn out to be equivalent to the continuum hypothesis instead.
I suppose that's what got Gamow.
Hmm, yes. My favourite example of a CH-equivalent problem is this (I was given the problem by Prof. Imre Leader):
"Can we colour the points in R^3 with the colours red, green and blue such that every line parallel to the x-axis contains only finitely many red points, every line parallel to the y-axis contains only finitely many green points, and every line parallel to the z-axis contains only finitely many blue points?"
Sincerely,
Adam P. Goucher
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