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An Infinity of Points

In response to:

Count Up from the December 3, 1987 issue

To the Editors:

…Martin Gardner, in his review of books by Eli Maor and Rudy Rucker [NYR, December 3, 1987], mistakenly says that “Cantor called the number that counts the real numbers (rational and irrational) aleph-one, or C,” and that “Cantor believed that 2 raised to the power of aleph-null is the same as C.” The latter proposition is not a “belief,” but a theorem of Cantor—who did call the number in question C, but not aleph-one. This last designation he applied to whatever cardinal number comes next in order after aleph-null; its identity with C, the number of real numbers or “power of the continuum,” is a famous conjecture of Cantor’s, his “continuum hypothesis.” Since it is now known (thanks to the work of Kurt Gödel in 1939 and Paul J. Cohen in 1963) that Cantor’s conjecture can neither be proved nor disproved on the basis of the principles generally accepted in mathematics, Gardner’s reference in his discussion of Rucker’s book to “the hand as an abstract solid with an aleph-one infinity of points” is incorrect….

Howard Stein

Chicago, Illinois

Martin Gardner replies:

Yes, the conjecture that aleph-one is the same as C (the power of continuum) is a theorem, but it was also Cantor’s passionate belief. Paul Cohen did indeed show that the theorem is undecidable by standard set theory. Assuming Cantorian set theory, a solid shape contains an aleph-one infinity of points. Assuming non-Cantorian set theory, the shape contains a C infinity of points. It would have been more accurate if I had said that the shape contains an uncountable infinity of points, but it is difficult to introduce these fine distinctions in compressed, nontechnical writing about mathematics.

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