Re: an important set theory post

On Thu, 24 Jul 2008 11:56:24 -0700 (PDT), calvin wrote:
On Jul 24, 1:56 pm, Dave Seaman <dsea...@xxxxxxxxxxxx> wrote:
On Thu, 24 Jul 2008 10:36:44 -0700 (PDT), calvin wrote:
On Jul 24, 1:07 pm, Dave Seaman <dsea...@xxxxxxxxxxxx> wrote:
On Thu, 24 Jul 2008 09:56:15 -0700 (PDT), calvin wrote:
Let me put it this way:
If n is the 'number of positive integers',

Then n is the cardinal aleph_0.

n + 1 = n
n + n = n
zn = n, for every positive integer z
Am I wrong?
Before recently I would have said that n = aleph0,
but now I'm so confused about what is ordinal and
what is cardinal, that I'm afraid to say it.

Who told you that the cardinality of the positive integers is not
aleph_0?
Nobody, but what I would have said, before all this
started, was that aleph_0 is the cardinality of any
set whose elements can be put into a one-to-one
correspondence with the positive integers; and I
would have said, informally, that 'the number of
positive integers' is aleph_0.

And you would have been correct.

But now I just don't know when to call something
ordinal or cardinal.  Omega has some relationship
to aleph_0 that I haven't grasped, probably
because they look so very much alike to me.

It's understandable that you can't grasp the difference, since the word
"order" does not seem to be in your vocabulary.

Two sets have the same cardinality if there is a bijection between them.
Two ordered sets have the same order type if there is an order
isomorphism (an order-preserving bijection) between them.

As long as you attempt to talk about ordinals without ever mentioning the
word "order", you will be perpetually confused.

In talking about the successor function I was talking about
order, but then I mistakenly thought that the successor
function was going to be the key to constructing aleph_1,
given aleph_0. But apparently the successor function can
be used to construct higher and higher ordinals, beyond
omega, but not higher cardinals beyond aleph_0.

There is an ordinal successor function, and there is a cardinal successor
function. They are different.

In any case, though, I'm well aware of the difference
between countable and uncountable, and that aleph_0
is the cardinality of such countable sets as the integers
and the rationals, but not the reals, whose cardinality
is c, arrived at by way of the bizarre notation, 2^aleph_0.
And I'm aware of the continuum hypothesis, that c=aleph_1,
which can be neither proven nor disproven.

But I'm still completely in the dark about how to
define, or construct, aleph_1. All I know about it
is that it belongs to the realm of the uncountable,
and is the next carninal greater than aleph_0.

The cardinal successor function is based on the Hartogs number, described
at <http://en.wikipedia.org/wiki/Hartogs_number>. Given any set X, there
is a least ordinal alpha = H(X) such that alpha cannot be injectively
mapped into X. This ordinal alpha is necessarily a cardinal, because it
cannot be bijectively mapped to any smaller ordinal.

If X happens to be a cardinal, then H(X) is the cardinal successor of X.


--
Dave Seaman
Third Circuit ignores precedent in Mumia Abu-Jamal ruling.
<http://www.indybay.org/newsitems/2008/03/29/18489281.php>
.

Relevant Pages

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