Definitions
from Wiktionary, Creative Commons Attribution/ShareAlike License
 n. The second of the transfinite cardinal numbers; according to the continuum hypothesis, it corresponds to the number of real numbers.
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Examples

Is the number of possible curves in space alephone, or a higher order of infinity?
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Assuming Cantorian set theory, a solid shape contains an alephone infinity of points.

Yes, the conjecture that alephone 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 alephone 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) alephone, or C," and that "Cantor believed that 2 raised to the power of alephnull 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 alephone.

As for the alephnull and alephone: it was proven that the continuum hypothesis essentially whether the cardinality of real numbers is alephone or higher is undecidable in standard set theory, so whether you want to accept it or not, you won’t hit any contradictions.
Intelligent Design explained: Part 2 random search  The Panda's Thumb
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