>John K Clark wrote:
>> Cantor proved that the number of integers, call it A, is the smallest
>> cardinal number. He proved that the amount of even numbers is the same as
>> the amount of all numbers. He proved that N*A =A if N is any finite
number,
>> and he even proved that A*A =A. However he also proved that 2^A is NOT
>> equal to A.
>>
>> Cantor's crowning achievement was when he proved something we now call
>> Cantor's Theorem, it states that if B is any cardinal number then B < 2^B.
>> This means there are an infinite number of cardinal numbers, an infinite
>> number of different infinities. However he was not able to figure out if
>> there is an infinite number between the number of integers and the number
of
>> points on a line, and even today it is not known.
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.
>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.