Re: [math-fun] Gamow syndrome (Aleph_0 and omega)
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 http://cp4space.wordpress.com
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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I contacted the (by then) publisher of "1, 2, 3 . . . Infinity" and let them know about these errors about 25 years ago, but as far as I can tell nothing has changed in subsequent printings. The worst thing is that a number of later writers just repeated Gamow's errors in their own books, without ever checking their accuracy with a mathematician. (These books include "Bridges to Infinity" by Michael Guillen, and "Infinity and More" by David Foster Wallach.) ----- Adam, I think you are being unduly charitable to suppose Gamow meant the number of ways you can color the plane with two colors (or in other words, the number of subsets of the plane) where he wrote "the number of geometrical curves". Every curve in the accompanying picture is continuous. ----- With the Axiom of Choice, cardinal numbers are indexed by the ordinals. Without the Axiom of Choice, things are apparently much murkier. In that case one can't even say that two sets are comparable. In fact, (under ZF set theory) AC is equivalent to the assertion that any two sets are comparable (i.e., one of them has a one-to-one map into the other). --Dan On 2012-12-07, at 4:40 AM, Adam P. Goucher 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').
On 12/7/12, Adam P. Goucher <apgoucher@gmx.com> wrote:
... 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?"
Well, I'll have to buy this one since nobody else has volunteered --- just how does the answer to this intriguingly elementary question depend on the continuum hypothesis? [I've a feeling that I once knew Imre Leader slightly, a long time ago when we were both postgraduate students; but can no longer recall where or how we met.] Fred Lunnon
participants (4)
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Adam P. Goucher -
Charles Greathouse -
Dan Asimov -
Fred lunnon