🚀 go-pugleaf

"Greg d. Moore" <mooregr@greenms.com> writes:

>	You both are.  Fundamentally you're doing the same thing, but in a
>different way.  You're describing all integers (negative and positive)
>he's dealling with integers and even integers.  Not a big difference. 
>All of them are equally infinite.

>Now the question is, there is as I understand it, at least a 3rd order
>of Infinity.  Does anyone know what it is?

There are in infinite number of orders of infinity. 

The lowest is aleph-null. This infinity is the size of the number of
even integers, which is the same as the number of odd integers, 
which is the same as the number of all integers, which is the same
as the number of rationals.

Another order of infinity (call is aleph-one) is the size of
all reals, and which is also the size of all sets of integer
numbers (including sets which contain aleph-null integers). 
It is also therefore the size of all aleph-null long sets
of rationals, or all aleph-null long sets of evens.

An even higher order of infinity is aleph-two. Now it's been a 
while since I've done this stuff, but I believe that this is
the size of all aleph-one long sets of reals. In other words,
sets of reals which can have as many elements in the set
as there are real numbers. One should not confuse this with
the set of all aleph-null long sets of reals -- that is, 
sets of reals with al many reals as integers. That turns
out to just be aleph-one again. 

One can continue this ad naseum -- i.e. aleph-n long sets of 
elements of an aleph-n size group gives aleph-(n+1). 

You can then make is sickeningly messy, making aleph-0,
aleph-1, aleph-2, aleph-3, aleph-4, and then, after (an
infinitely long time) making aleph-(aleph-null). And then
aleph-(aleph-1), and aleph-(aleph-2), and aleph-(aleph-3),
and after a very long time aleph-(aleph-(aleph-null)).
And eventually aleph-(aleph-(aleph-(aleph-null))), and then
aleph-(aleph-(aleph-(aleph-(aleph-null)))). And then this statement,
with a countable number of "aleph-"'s written down, which is 
believe is called w (that's supposed to be an omega).
Then you can repeat this whole mess with the w's. All these
levels of infinity are different.

I may be messing up the definitions a bit. I think, strictly speaking,
aleph-one may refer to the ""smallest" set of infinity which is
larger than the set of integers. The reals are larger than the
integers, but it is not known if there is a size of infinity 
between the integers and the reals. Similarly, it's not known
if there are levels of  infinity between the ones generated by the
process described above. I think it might be one of these
Godel-like things. 

I know it probably sounds like I'm making this up, but it's really
true. It's been a while since I've looked at this stuff, but I'm
pretty sure I at least have the gist of the process right. I'll
check it up when I get home.

Momo