Re: an important set theory post

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

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.

(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, 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.
.

Relevant Pages

  • Re: cardinality multiplication : a*b = max(a,b)
    ... infinite, then ab = max. ... whose domain is XxX for some subset X of A and which are bijections ... with domain X union Y that extends f, ... XxY, YxX, all of which have the same cardinality as X, so there are ...
    (sci.math)
  • Re: connected subsets of R^2
    ... >>cardinality of the continuum, so well-order F union G by the first ... non-empty and disjoint) Z has card. ... Cardinality of the continuum, and their intersections ...
    (sci.math)
  • Re: The definition of finite set
    ... which I think can be modelled as the cardinality of the set ... Without using the axiom of choice, how would you prove that the union ... of a countably infinite collection of unordered pairs has a countably ...
    (sci.logic)
  • Re: cardinality multiplication : a*b = max(a,b)
    ... Derek Holt wrote: ... infinite, then ab = max. ... with domain X union Y that extends f, ... XxY, YxX, all of which have the same cardinality as X, so there are ...
    (sci.math)
  • Re: an important set theory post
    ... is a real mathematician and who is a crank. ... David dares to call me timmy instead of tommy, and he calls me a crank for USING words HE DOESNT KNOW. ... C is called the cardinality of S.) ... the union of aleph_o' for all o' < o. ...
    (sci.math)
Quantcast