Re: Infinities [was Re: The Big Bang]
Wed, 5 Nov 1997 19:09:42 -0500 (EST)

In a message dated 11/5/97 5:07:16 AM, wrote:

>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
>> 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
>> 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
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
that the number of points on a line = aleph-one = 2^(aleph-null), just
it makes the math easier. This is called the "Continuum Hypothesis" (or

That the Continuum Hypothesis can be assumed to be either true or false
trouble seems weird. I conceptualize it by thinking that the number of
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
and the line. If you deny it, then aleph-one, aleph-two, etc. all come in

>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
>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
hypothesis" which holds that the number of all possible functions is
I believe it's also undecidable but generally used because it makes life