From: Mitchell Porter (mitch@thehub.com.au)
Date: Wed Nov 05 1997 - 18:25:19 MST
[Curt Adams]
> It's been shown that you can assume either that the number of points on the
> line
> is aleph-one or aleph-infinity and you'll get no contradiction. It's much
> like Euclidean vs. non-Euclidean geometry. Most people work with the
> assumption
> that the number of points on a line = aleph-one = 2^(aleph-null), just
> because
> it makes the math easier. This is called the "Continuum Hypothesis" (or
> axiom).
>
> That the Continuum Hypothesis can be assumed to be either true or false
> without
> trouble seems weird. I conceptualize it by thinking that the number of
> points
> on a line is an ill-defined thing.
Just to clarify: what's been proved is that the Continuum Hypothesis
is independent of the ordinary (Zermelo-Fraenkel) axioms of set theory.
This is done by constructing "models", universes of sets which satisfy
the ZF axioms and which also satisfy whatever postulate regarding the
cardinality of the continuum you've chosen to work with.
How to interpret this situation is not clear. Some would see it as
evidence that the transfinite is just a product of the human
imagination. Goedel, I think, felt that it simply meant that the
current axiomatizations of set theory are incomplete, and that by
further thought and clarification of concepts we would eventually
see what the extra axioms should be.
> >By this do you mean that there are B integers and 2^B points on a line, and
> >we don't know if there's anything in between?
>
> I don't think there's any cardinality between aleph-null and aleph-one.
> If you assume the Continuum Hypothesis then there's nothing between the
> integers
> and the line. If you deny it, then aleph-one, aleph-two, etc. all come in
> between.
>
> >Also, where do aleph-null, aleph-one, etc. fit into all this? My dictionary
> >says that aleph-null is "the first transfinite number". Is that what you
> call
> >A? If so, is aleph-one what you call 2^A?
>
> Yes. Aleph-two is 2^(aleph-one) and so on. There's also an "extended
> continuum
> hypothesis" which holds that the number of all possible functions is
> aleph-two.
> I believe it's also undecidable but generally used because it makes life
> easier.
My understanding: the Continuum Hypothesis is 2^(aleph-null) = aleph-one.
The Generalized Continuum Hypothesis is that 2^(aleph-X) = aleph-(X+1).
There is no cardinality between aleph-null and aleph-one by definition;
aleph-one is *not* defined as the number of points on the real line
("the cardinality of the continuum"), but simply as "the next biggest
cardinal after aleph-null". I think the original demonstration of
a model of ZF that contradicted the CH (Philip Cohen, 1963?) *can*
work with cardinals of the form aleph-N, where 0 < N < infinity.
Finally, what one *can* say about the number of all functions (from
the reals onto the reals) is not that it's aleph-two, but that it's
2^(2^(aleph-null)).
-mitch
http://www.thehub.com.au/~mitch
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