Wednesday, May 25, 2005

CH in simple terms

CH is an abbreviation for the continuum hypothesis. Taking my audience into account, I'm not going to formally define CH here. Instead, I'm going to present a statement equivalent to CH yet expressable in terms of high school (middle school?) level math.

Picture three-dimensional (Euclidean) space. For reference, add an x-axis, a y-axis, and a z-axis to the picture. (My usual picture is the x-axis running from left to right, the y-axis running from front to back, the z-axis running from bottom to top.) Now imagine trying to assign to every point in this space one of three colors: red, green, or blue. Too easy? Well, not just any coloring will do. There are three requirements:

1) Every line parallel to the x-axis contains at most finitely many red points.
2) Every line parallel to the y-axis contains at most finitely many green points.
3) Every line parallel to the z-axis contains at most finitely many blue points.

Is there such a strange coloring? The answer is "yes" if and only if CH is true.

If you don't like trying to color things of infinite length, then just try to color all the points inside a cube (or any other solid shape) so as to meet the above three requirements. The existence of such a coloring is still equivalent to CH.

The above equivalent form of CH is originally due to Sierpinski (cite: Simms, John C.; Sierpinski's theorem. Simon Stevin 65 (1991), no. 1-2, 69--163).

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