🚀 go-pugleaf

In article <momo.861569516@sbphy.physics.ucsb.edu>, Monwhea Jeng
<momo@sbphy.physics.ucsb.edu> writes
>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.

Aleph-2 is also the number of curves that can be drawn on the plane. It
is, IIRC, the last infinity that has some non-trivial meaning.

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

aleph-(n+1) equals 2 to the power aleph-n.

>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).

No, w is the first "ordinal" infinity. This is, as I understand it, when
you distinguish the order of the items in an infinite set:

    w:     1, 2, 3, 4, 5, 6 ...
    w+1:   2, 3, 4, 5, 6, ... 1
    w+2:   3, 4, 5, 6, ... 1, 2
    w+3:   4, 5, 6, ... 1, 2, 3
    2w:    1, 3, 5, ... 2, 4, 6, ...
    2w+2:  2, 4, 6, ... 3, 5, 7, ... 1

>I think, strictly speaking,
>aleph-one may refer to the ""smallest" set of infinity which is
>larger than the set of integers.

No. aleph-1 is 2 to the power aleph-0, and it is possible to prove that
that is the number of real numbers.

>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 recall an article that said that it is actually possible to prove that
you can't prove it either way - in other words, the statements:
- ordinary arithmetic is correct, and there is a number between aleph-0
  and aleph-1;
- ordinary arithmetic is correct, and there are no numbers between
  aleph-0 and aleph-1;
both lead to consistent systems.

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