aleph-one

Is the number of possible curves in space aleph-one, or a higher order of infinity?

A brief look at two comments on one ID-creo site - The Panda's Thumb 2005

Is the number of possible curves in space aleph-one, or a higher order of infinity?

A brief look at two comments on one ID-creo site - The Panda's Thumb 2005

Assuming Cantorian set theory, a solid shape contains an aleph-one infinity of points.

An Infinity of Points Stein, Howard 1988

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.

An Infinity of Points Stein, Howard 1988

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….

An Infinity of Points Stein, Howard 1988

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."

An Infinity of Points Stein, Howard 1988

The latter proposition is not a "belief," but a theorem of Cantor — who did call the number in question C, but not aleph-one.

An Infinity of Points Stein, Howard 1988

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.

Intelligent Design explained: Part 2 random search - The Panda's Thumb 2006