From: Hal Finney (hal@rain.org)
Date: Sun Nov 09 1997 - 11:20:09 MST
John Clark, <johnkc@well.com>, writes:
> On Sat, 8 Nov 1997 Hal Finney <hal@rain.org> Wrote:
> >No, Pw has been proven to have larger cardinality than aleph null,
> >and it has been proven to have the same cardinality as the real
> >numbers. But it is not known whether it is equal to aleph one. [...]
> >Show me such a proof!
>
> Page 238 of "Infinity and the Mind" by Rudy Rucker.
There is no such proof on that page. What I do find is:
"CH is the statement 2^aleph-null = aleph-one."
"...look first at a number of different sets that have the cardinality
2^aleph-null: the power set of omega,... the unit interval..."
Clearly he is not proving that Pw (the power set of omega) has the
cardinality of aleph-one. He states that Pw has the same cardinality as
2^aleph-null, and he says that the CH says that 2^aleph-null = aleph-one.
So if he were proving what you say, he would be claiming to prove the CH,
which he says is impossible.
Note too that he clearly says that the cardinality of 2^aleph-null,
Pw, and the unit line are the same.
I'm sorry to be dragging this issue out so much, but I'm baffled that
you could be reading the same book that I am and reach such different
conclusions.
In addition to the other URLs I gave, you might check the sci.math FAQ,
which discusses CH. One location is:
http://daisy.uwaterloo.ca/~alopez-o/math-faq/node38.html
This (which apparently was written by David Chalmers) leans towards the
view that CH is false, that the cardinality of Pw and C is much greater
than aleph-one or -two. (Some people think you'd have to go through an
infinite number of alephs before you get to it, possibly an uncountably
infinite number of them!) This FAQ also talks about the controversy
over whether CH can even be said to have a truth value at all. There are
many schools of thought about this issue among mathematicians, which IMO
is another aspect of mathematical research which doesn't fit that well
into what I understand of Penrose's model.
There is also a Continuum Hypothesis FAQ at:
http://www.ii.com/math/ch/faq/
This opens:
> The continuum hypothesis was proposed by Georg Cantor in 1877
> after he showed that the real numbers cannot be put into one-to-one
> correspondence with the natural numbers. Cantor hypothesized that the
> number of real numbers is the next level of infinity above the number of
> natural numbers. He used the Hebrew letter aleph to name the different
> levels of infinity: aleph_0 is the number of (or cardinality of) the
> natural numbers or any countably infinite set, and the next levels of
> infinity are aleph_1, aleph_2, aleph_3, etc. Since the reals form the
> quintessential continuum, Cantor named the cardinality of the reals c,
> for continuum. Cantor's original formulation of the continuum hypothesis,
> or CH, can be stated as either:
>
> 1.card(R)=aleph_1
> 2.c=aleph_1
>
> where `card(R)' means `the cardinality of the reals.' An amazing fact
> that Cantor also proved is that the cardinality of the set of all subsets
> of the natural numbers -- the power set of N or P(N) -- is equal to the
> cardinality of the reals.
Hal
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