Re: an important set theory post

On Wed, 23 Jul 2008 07:19:24 -0700 (PDT), calvin
<crice5@xxxxxxxxxxxxxx> wrote:

On Jul 23, 7:49 am, David C. Ullrich <dullr...@xxxxxxxxxxx> wrote:
The facts:
First, define "A < B" if there is an injection
from A to B but none from B to A.

Yes, that has always been my understanding, too.

Now, by definition a cardinal is an ordinal
A such that B < A for all previous ordinals B.

Okay, but the definition I was given in colledge
was that a cardinal is a 'size'. Not an actual
number, but a 'size' relative to other cardinals.

That's a reasonable explanation to give to kids
who aren't prepared to learn that actual math.
But it's totally unsuitable as a _definition_;
it just replaces the question "what is a cardinal"
by "what do you mean by 'a size', exactly?".

I fully understand why the 'size', aleph_0, is less
than the 'size', C, for example.

That requires proof, because C (usually written
c) is not defined to be this aleph or that aleph.

Aleph_0 is the
'size' of any set that can be mapped one-to-one
with the natural numbers (which do not include 0,
by the way.) C is the 'size' of any set that can
be mapped one-to-one with the real numbers. And,
yes, aleph_0 is the smallest 'size' of an infinite
set, as my understanding has been.

Now, you say that real mathematicians have
corrected me. But I have no way of knowing who
is a real mathematician and who is a crank.

The rest of us have no problem with that.
You can tell by whether or not almost everything
someone says results in numerous corrections
from numerous parties.

The reason I first posted here was because
someone used the word, 'subcountable', and used
it in a non-grammatical way, so I thought he
probably was a crank. And he was using non-integer
aleph subscripts; so I tried to correct him,
and things developed from there.

Timmy is about aa cranky as they come.
That has no bearing on the fact that your
"correction" was simply wrong.

I gather that you are a real mathematician, so I
am dismayed that you continue to insist that a
'size' is a set. There are many popularized
math books out there, that are forever saying
nonsense things about infinity, talking like a
sideways 8 is an actual number, for example. It
is impossible to tell who is who, so all I can
do it try to hold on to the understanding that
I acquired in college, which is at least a little
better than what crank math popularizers seem to
have.

So. You learned something wrong in college.
And now you have no choice but to cling to
that error. Right.

(Using AC we can show that if S is a set then
there exists a cardinal C such that S is equinumerous
with C (ie there is a bijection between the two).
C is called the cardinality of S.)

Sounds good to me, though I don't know what AC
means, other than 'alternating current'.

Now, if C is a cardinal then there is a smallest
cardinal C' larger than C. Also, if S is any set
of cardinals it's easy to show that the union
of the elements of S is a cardinal.

Yes, I also learned in college that for every
carninal there is a next larger cardinal. I don't
remember the proof, however.

So we define aleph_o for all ordinals o as follows:

(i) aleph_0 is the smallest infinite cardinal.

Why not define it directly, as the cardinal of
any set that can be mapped one-to-one with the
natural numbers?

(ii) If o' is the successor of o then aleph_o'
is the smallest cardinal greater than aleph_o.

Fine.

(iii) If o is a limit ordinal then aleph_o is
the union of aleph_o' for all o' < o.

There you go again, treating cardinals like
sets,

That's because cardinals _are_ sets.

You're really becoming totally unreasonable.
You admit you don't really understand this
and that, all you know is what you were told
in college many years ago, but you continue
to argue when people give you the correct
_definitions_. A cardinal is in fact a certain
sort of set - whether you believe that or not
has no bearing on the facts of the matter.

and forming unions, which I believe are
defined only for sets. There is no union
of 5 and 6, but there is a union of a set that
contains the element 5 and a set that contains
the element 6, and that union contains the
elements 5 and 6.

That gives us a well-defined aleph_o for
every ordinal o. And in fact the union of
aleph_n for natural numbers n is exactly
equal to aleph_omega, by (iii).

If that's the way real mathematicians want it,
I guess that's the way it has to be, but I wish
they had come up with a new notation for a
'size' of a set that is the union of infinitely many
sets, each having the least cardinality greater
than the previous, starting with a set of
lowest infinite cardinality, ie. cardinality
aleph_0.

David C. Ullrich

"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
.

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