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Topic: Another AC anomaly?
Replies: 280   Last Post: Dec 27, 2009 3:53 PM

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anonymous.rubbertube@yahoo.com

Posts: 245
Registered: 10/20/08
Re: Another AC anomaly?
Posted: Dec 8, 2009 1:36 PM
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On Dec 8, 11:10 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 8 Dez., 16:13, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
>

> >  > Nevertheless it is a limit ordinal.
>
> > Yes, that does not mean that necessarily a limit is involved.  It is a limit
> > in the sense that you do not get there by continuously getting at the
> > successor, in that case it is a limiting process.  But when you define
> > N as an infinite union you do *not* go there by continuously getting at
> > a successor.  The union of a collection (finite, countably infinite or
> > some other infinity) is defined whithout resorting to successor operations.
> > Moreover, they would even not make sens if the collection is infinite but
> > not countably infinite.

>
> That does not make sense in either respect, so or so.
>
>
>

> >  > As a starting point, we use the fact hat each natural number is
> >  > identified with the set of all smaller natural numbers: n = {m in N :
> >  > m < n}.

>
> > Note that here N is apparently already defined, without using a limit.
>
> Natural numbers can be defined without using a set.
>
>
>

> >  >         Thus we let w, the least transfinite number, to be the set N
> >  > of all natural numbers: w = N = {0, 1, 2, 3, ...}.
> >  >   It is easy to continue the process after this 'limit' step is made:
> >  > The operation of successor can be used to produce numbers following w
> >  > in the same way we used it to produce numbers following 0.

>
> > Yes.  So what?  That you can define things using a limit does *not*
> > imply that it is necessarily defined as a limit.

>
> O I see. That's like cardinality. The limit cardinality is not the
> cardinality of the limit (because the limit is not a limit).
>
> Regards, WM



A function f is said to be continuous at a point x in its domain if
the limit of f(a), as a approaches x, is equal to f(x); in others
words, the limit of the values of f is equal to the value of f at the
limit, speaking loosely. Of course, not every function is continuous
at every point in its domain, and some functions are not even
continuous at any point in their domains at all.

The situation for sets and cardinality is no more mysterious than
that. The cardinality of a limit of subsets of the integers is not
guaranteed to be the limit of the cardinalities of those subsets. You
don't expect an arbitrary function to always be continuous, so perhaps
it's unreasonable to expect the cardinality "function," defined on
subsets of the integers, to be continuous.


Date Subject Author
11/23/09
Read Re: Another AC anomaly?
Jesse F. Hughes
11/23/09
Read Re: Another AC anomaly?
Herman Jurjus
11/23/09
Read Re: Another AC anomaly?
master1729
11/25/09
Read Re: Another AC anomaly?
T.H. Ray
11/24/09
Read Re: Another AC anomaly?
george
12/1/09
Read Re: Another AC anomaly?
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11/25/09
Read Re: Another AC anomaly?
Bill Taylor
11/26/09
Read Re: Another AC anomaly?
Daryl McCullough
11/30/09
Read What's wrong with supertasks? (was Re: Another AC anomaly?)
Herman Jurjus
12/1/09
Read Jesse Hughes of Bentley College uses signature block to defame
others; Ann Reeves Re: What's wrong with supertasks?
plutonium.archimedes@gmail.com
12/1/09
Read Re: Jesse Hughes of Bentley College uses signature block to defame
others; Ann Reeves Re: What's wrong with supertasks?
Marshall
12/1/09
Read using signature-blocks to defame innocent people Re: Jesse Hughes of
Bentley College
plutonium.archimedes@gmail.com
12/1/09
Read Re: Jesse Hughes of Bentley College uses signature block to defame
others; Ann Reeves Re: What's wrong with supertasks?
Seth Breidbart
12/1/09
Read Re: using signature-blocks to defame innocent people Re: Jesse Hughes
of Bentley College
Marshall
12/2/09
Read Re: using signature-blocks to defame innocent people Re: Jesse Hughes
of Bentley College
Marshall
11/27/09
Read Re: Another AC anomaly?
William Hughes
11/27/09
Read Re: Another AC anomaly?
William Hughes
11/26/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/28/09
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mueckenh@rz.fh-augsburg.de
11/28/09
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Virgil
11/26/09
Read Re: Another AC anomaly?
William Hughes
11/28/09
Read Re: Another AC anomaly?
William Hughes
11/26/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/28/09
Read Re: Another AC anomaly?
ross.finlayson@gmail.com
11/28/09
Read Re: Another AC anomaly?
Virgil
11/29/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/29/09
Read Re: Another AC anomaly?
Virgil
11/29/09
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ross.finlayson@gmail.com
11/29/09
Read Re: Another AC anomaly?
Marshall
11/30/09
Read Re: Another AC anomaly?
Virgil
11/30/09
Read Re: Another AC anomaly?
ross.finlayson@gmail.com
11/30/09
Read Re: Another AC anomaly?
ross.finlayson@gmail.com
11/30/09
Read Re: Another AC anomaly?
Virgil
11/30/09
Read Re: Another AC anomaly?
ross.finlayson@gmail.com
11/30/09
Read Re: Another AC anomaly?
Virgil
11/30/09
Read Re: Another AC anomaly?
ross.finlayson@gmail.com
11/30/09
Read Re: Another AC anomaly?
Virgil
11/29/09
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mueckenh@rz.fh-augsburg.de
11/26/09
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LauLuna
11/26/09
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William Hughes
11/26/09
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anonymous.rubbertube@yahoo.com
11/29/09
Read Re: Another AC anomaly?
William Hughes
11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
Virgil
11/27/09
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11/29/09
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William Hughes
11/30/09
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mueckenh@rz.fh-augsburg.de
11/30/09
Read Re: Another AC anomaly?
Virgil
11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
Virgil
11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
Virgil
11/27/09
Read Re: Another AC anomaly?
William Hughes
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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11/27/09
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Virgil
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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Dik T. Winter
11/27/09
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mueckenh@rz.fh-augsburg.de
11/30/09
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Dik T. Winter
11/30/09
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mueckenh@rz.fh-augsburg.de
11/30/09
Read Re: Another AC anomaly?
Virgil
12/1/09
Read Re: Another AC anomaly?
Dik T. Winter
12/1/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/1/09
Read Re: Another AC anomaly?
Dik T. Winter
12/1/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/1/09
Read Re: Another AC anomaly?
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12/1/09
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Virgil
12/1/09
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mueckenh@rz.fh-augsburg.de
12/1/09
Read Re: Another AC anomaly?
Virgil
12/1/09
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ross.finlayson@gmail.com
12/1/09
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Virgil
12/1/09
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ross.finlayson@gmail.com
12/1/09
Read Re: Another AC anomaly?
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12/2/09
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12/2/09
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mueckenh@rz.fh-augsburg.de
12/2/09
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12/2/09
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12/3/09
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12/3/09
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12/3/09
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12/7/09
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12/9/09
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12/11/09
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12/15/09
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12/19/09
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12/19/09
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12/22/09
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12/22/09
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12/18/09
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12/19/09
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12/21/09
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12/21/09
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12/27/09
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12/16/09
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12/11/09
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12/11/09
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12/11/09
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12/11/09
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12/12/09
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12/10/09
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12/3/09
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12/3/09
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12/7/09
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12/8/09
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12/8/09
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12/8/09
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12/10/09
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12/10/09
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12/10/09
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12/10/09
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12/11/09
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12/11/09
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12/11/09
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12/15/09
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12/16/09
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12/16/09
Read Re: Another AC anomaly?
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12/16/09
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mueckenh@rz.fh-augsburg.de
12/17/09
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12/17/09
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12/4/09
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mueckenh@rz.fh-augsburg.de
12/4/09
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12/4/09
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12/7/09
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Dik T. Winter
12/7/09
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mueckenh@rz.fh-augsburg.de
12/7/09
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12/8/09
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12/8/09
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mueckenh@rz.fh-augsburg.de
12/8/09
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12/10/09
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12/10/09
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12/10/09
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12/10/09
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12/11/09
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12/11/09
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12/11/09
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12/12/09
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12/19/09
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12/27/09
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12/27/09
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12/13/09
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12/4/09
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12/4/09
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12/4/09
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12/4/09
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12/4/09
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12/5/09
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12/5/09
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12/5/09
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12/6/09
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12/5/09
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12/2/09
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12/1/09
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12/1/09
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11/27/09
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Virgil
11/27/09
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anonymous.rubbertube@yahoo.com
11/30/09
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William Hughes
11/27/09
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William Hughes
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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11/27/09
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anonymous.rubbertube@yahoo.com
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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Virgil
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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Virgil
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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William Hughes
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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Virgil

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