from Wiktionary, Creative Commons Attribution/Share-Alike License
- n. The second of the transfinite cardinal numbers; according to the continuum hypothesis, it corresponds to the number of real numbers.
Sorry, no etymologies found.
Is the number of possible curves in space aleph-one, or a higher order of infinity?
Assuming Cantorian set theory, a solid shape contains an aleph-one infinity of points.
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.
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 .
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.
As for the aleph-null and aleph-one: it was proven that the continuum hypothesis essentially whether the cardinality of real numbers is aleph-one or higher is undecidable in standard set theory, so whether you want to accept it or not, you won’t hit any contradictions.
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