This is not quite right; it is easy to show that 2^B equals C, where B
is aleph-null. Simply identify the points on the unit segment with their
representation in base two. The number of possible binary representations
of length n is 2^n, so the total number of binary representations of
length aleph-null is 2^aleph-null, and that is the number of points on
the line. (This is a slight oversimplification but it is the basic idea.)
> Even today the "Continuum Hypothesis" which states that C is
> equal to alpha-one (2^B) has never been proven and remains of the great
> mysteries of mathematics.
The Continuum Hypothesis does say that C is equal to aleph-one, but the
issue in question is what is aleph-one. There are no sets known to
have the cardinality of aleph-one, as I understand it. Aleph-one is
defined as the smallest cardinal greater than aleph-zero.
> Cantor also showed that C^C = F > C where F is the number of all one valued
> functions, there are more curves in the plane than there are points.
> It's known that F is one of the alpha numbers (alpha -two?) but it's not
> known which one.
The Continuum Hypothesis (CH) is known to be undecideable under current
axioms of set theory. Here is what Rudy Rucker says about possible
resolutions, from Infinity and the Mind:
"In the late 1960s, Kurt Godel did suggest some new axioms that would
decide the size of C. These were the so-called omega-square and
aleph-one-square axioms.... At first it was thought that these two
axioms, plus one other, implied that C = aleph-two. But then Gaisi
Takeuti showed that on the basis of the two square axioms alone, one
can prove that C = aleph-one. At present, there are no very popular
axioms that imply that C is not equal to aleph-one, so it may be that
CH will come to be accepted in the future.
"One might hope that simply by thinking about C and aleph-one, one could
decide whether or not CH is plausible. In 1934, Waclaw Sierpinski did
womething like this in assembling a book of eighty-eight statements
equivalent to CH. But none of these statements is obviously true or
false - although in his famous 1947 essay, 'What is Cantor's Continuum
Problem?,' Godel claimed that some of the equivalents of the continuum
hypothesis were 'highly implausible.'"
This is something they don't talk about much in math class - what do
you do when your axioms have failed you and can't decide a question?
What kind of reasoning works, how do you approach the problem? It is
related to such issues as Penrose's "proof" that Godel's theorem implies
that machines cannot think.
The question is whether the task of finding new axioms is itself one
which can be described formally and axiomatically, or equivalently,
one which can be captured in a computer program. Or do we somehow
have a non-axiomatic component which leads us to truth, as Penrose
believes? This problem might be a good test case. It sounds like the
preference is to set C = aleph-one, although maybe Godel was pushing
for C = aleph-two. Would Penrose claim that we have an infallible
non-axiomatic intuition which guarantees that we make the right choice?
I don't understand his argument well enough to know what he would say.
Hal