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

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mueckenh@rz.fh-augsburg.de

Posts: 18,076
Registered: 1/29/05
Re: Another AC anomaly?
Posted: Dec 7, 2009 11:34 AM
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On 7 Dez., 16:37, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> In article <yvCdnW28VrXBqIXWnZ2dnUVZ_s-dn...@giganews.com> "K_h" <KHol...@SX729.com> writes:
>
> ...
>  > > > > When you mean with your statement about N:
>  > > > >      N = union{n is natural} {n}
>  > > > > then that is not a limit.  Check the definitions about
>  > > > > it.
>  > > >
>  > > > It is a limit. That is independent from any definition.
>  > >
>  > > It is not a limit.  Nowhere in the definition of that
>  > > union a limit is used
>  > > or mentioned.
>  >
>  > Question.  Isn't this simply a question of language?
>
> Not at all.  When you define N as an infinite union there is no limit
> involved, there is even no sequence involved.  N follows immediately
> from the axioms.


Nevertheless it is a limit ordinal.
>
>  >                                                       My
>  > book on set theory defines omega, w, as follows:
>  >
>  >     Define w to be the set N of natural numbers with its
>  > usual order
>  >     < (given by membership in ZF).
>  >
>  > Now w is a limit ordinal so the ordered set N is, in the
>  > ordinal sense, a limit.  Of course w is not a member of N
>  > becasuse then N would be a member of itself (not allowed by
>  > foundation).
>
> Note here that N (the set of natural numbers) is *not* defined using a
> limit at all.  That w is called a limit ordinal is a definition of the
> term "limit ordinal".  It does not mean that the definition you use to
> define it actually uses a limit.  (And if I remember right, a limit
> ordinal is an ordinal that has no predecessor, see, again no limit
> involved.)


Karel Hrbacek and Thomas Jech: "Introduction to Set Theory"
Marcel Dekker Inc., New York, 1984, 2nd edition, p. 134

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}. 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:

Regards, WM


Date Subject Author
11/23/09
Read Re: Another AC anomaly?
Jesse F. Hughes
11/23/09
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Herman Jurjus
11/23/09
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master1729
11/25/09
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T.H. Ray
11/24/09
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george
12/1/09
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11/25/09
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Bill Taylor
11/26/09
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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
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plutonium.archimedes@gmail.com
12/1/09
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others; Ann Reeves Re: What's wrong with supertasks?
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12/1/09
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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
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William Hughes
11/26/09
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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
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William Hughes
11/28/09
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William Hughes
11/26/09
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mueckenh@rz.fh-augsburg.de
11/28/09
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ross.finlayson@gmail.com
11/28/09
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Virgil
11/29/09
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mueckenh@rz.fh-augsburg.de
11/29/09
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Virgil
11/29/09
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ross.finlayson@gmail.com
11/29/09
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Marshall
11/30/09
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Virgil
11/30/09
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ross.finlayson@gmail.com
11/30/09
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ross.finlayson@gmail.com
11/30/09
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Virgil
11/30/09
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ross.finlayson@gmail.com
11/30/09
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Virgil
11/30/09
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ross.finlayson@gmail.com
11/30/09
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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
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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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Virgil
11/27/09
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Alan Smaill
11/29/09
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11/30/09
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mueckenh@rz.fh-augsburg.de
11/30/09
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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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Virgil
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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Alan Smaill
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
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Virgil
12/1/09
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Dik T. Winter
12/1/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/1/09
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Dik T. Winter
12/1/09
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mueckenh@rz.fh-augsburg.de
12/1/09
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William Hughes
12/1/09
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Virgil
12/1/09
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mueckenh@rz.fh-augsburg.de
12/1/09
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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
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Virgil
12/2/09
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Dik T. Winter
12/2/09
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mueckenh@rz.fh-augsburg.de
12/2/09
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Dik T. Winter
12/2/09
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mueckenh@rz.fh-augsburg.de
12/2/09
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12/3/09
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12/3/09
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mueckenh@rz.fh-augsburg.de
12/3/09
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Dik T. Winter
12/3/09
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12/7/09
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12/7/09
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Virgil
12/8/09
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12/8/09
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Virgil
12/9/09
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12/9/09
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12/9/09
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12/9/09
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12/9/09
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12/10/09
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12/9/09
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12/9/09
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12/10/09
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12/10/09
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mueckenh@rz.fh-augsburg.de
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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12/15/09
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12/15/09
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mueckenh@rz.fh-augsburg.de
12/15/09
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12/15/09
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12/16/09
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12/16/09
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12/16/09
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12/17/09
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12/17/09
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12/17/09
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12/17/09
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12/17/09
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12/17/09
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12/18/09
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12/18/09
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12/18/09
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12/19/09
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mueckenh@rz.fh-augsburg.de
12/19/09
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12/22/09
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12/22/09
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Virgil
12/18/09
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Dik T. Winter
12/19/09
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mueckenh@rz.fh-augsburg.de
12/19/09
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12/21/09
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Dik T. Winter
12/21/09
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mueckenh@rz.fh-augsburg.de
12/21/09
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12/21/09
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12/22/09
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Dik T. Winter
12/27/09
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mueckenh@rz.fh-augsburg.de
12/27/09
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12/16/09
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mueckenh@rz.fh-augsburg.de
12/16/09
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Virgil
12/11/09
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K_h
12/11/09
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Dik T. Winter
12/11/09
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K_h
12/11/09
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12/12/09
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12/12/09
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12/12/09
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12/11/09
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mueckenh@rz.fh-augsburg.de
12/11/09
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12/10/09
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mueckenh@rz.fh-augsburg.de
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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12/10/09
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Dik T. Winter
12/11/09
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12/11/09
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12/7/09
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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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12/8/09
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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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12/3/09
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12/3/09
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mueckenh@rz.fh-augsburg.de
12/3/09
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12/7/09
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Dik T. Winter
12/7/09
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Virgil
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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Dik T. Winter
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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Dik T. Winter
12/10/09
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mueckenh@rz.fh-augsburg.de
12/10/09
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12/10/09
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Dik T. Winter
12/11/09
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mueckenh@rz.fh-augsburg.de
12/11/09
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12/11/09
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12/15/09
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Dik T. Winter
12/16/09
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mueckenh@rz.fh-augsburg.de
12/16/09
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Dik T. Winter
12/16/09
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mueckenh@rz.fh-augsburg.de
12/17/09
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T.H. Ray
12/17/09
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Dik T. Winter
12/4/09
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mueckenh@rz.fh-augsburg.de
12/4/09
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Virgil
12/4/09
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Marshall
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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Virgil
12/8/09
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Dik T. Winter
12/8/09
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mueckenh@rz.fh-augsburg.de
12/8/09
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Virgil
12/10/09
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Dik T. Winter
12/10/09
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mueckenh@rz.fh-augsburg.de
12/10/09
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Virgil
12/10/09
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Dik T. Winter
12/11/09
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mueckenh@rz.fh-augsburg.de
12/11/09
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12/11/09
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mueckenh@rz.fh-augsburg.de
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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mueckenh@rz.fh-augsburg.de
12/12/09
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12/12/09
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12/12/09
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12/12/09
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12/12/09
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12/12/09
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12/12/09
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12/12/09
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12/13/09
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12/16/09
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12/18/09
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12/19/09
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12/19/09
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12/19/09
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12/19/09
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12/19/09
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Virgil
12/19/09
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Virgil
12/19/09
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ross.finlayson@gmail.com
12/19/09
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12/22/09
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ross.finlayson@gmail.com
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12/27/09
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12/21/09
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12/21/09
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12/22/09
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12/27/09
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mueckenh@rz.fh-augsburg.de
12/27/09
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12/27/09
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12/27/09
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Virgil
12/27/09
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12/27/09
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12/13/09
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mueckenh@rz.fh-augsburg.de
12/13/09
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Virgil
12/4/09
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K_h
12/4/09
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mueckenh@rz.fh-augsburg.de
12/4/09
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Virgil
12/4/09
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mueckenh@rz.fh-augsburg.de
12/4/09
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Virgil
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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Virgil
12/5/09
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Carsten Schultz
12/2/09
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12/1/09
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george
12/1/09
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Virgil
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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William Hughes
11/27/09
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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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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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