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

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Dik T. Winter

Posts: 7,899
Registered: 12/6/04
Re: Another AC anomaly?
Posted: Dec 10, 2009 9:06 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

In article <Virgil-1E8B09.00355309122009@newsfarm.iad.highwinds-media.com> Virgil <Virgil@home.esc> writes:
> In article <c8idnc6JG6hv34LWnZ2dnUVZ_q-dnZ2d@giganews.com>,
> "K_h" <KHolmes@SX729.com> wrote:

...
> > > In ZF, unions are defined only for sets of sets and for
> > > such a set of
> > > sets S, the union is defined a the set of all elements of
> > > elements of S.

> >
> > Check out:
> > http://planetmath.org/encyclopedia/SequenceOfSetsConvergence.html


With that definition set inclusion is a requirement, with the definition
I gave it is not.

> > - Let N be the limit set formed from the initial set {}.
> >
> > In this case N is a convergent set:
> >
> > http://planetmath.org/encyclopedia/SequenceOfSetsConvergence.html

>
> However, in the discussion between Dik and WM, Dik gave SPECIFIC
> definition of what HE meant by the limit of a seqeunce of sets which
> differed from that in your citation.


The difference is that with my definition (which is also quite generally
used) sequences of sets can also converge if there is no set inclusion,
as in the sequence {n}. If A_n subset A_(n+1) or A_(n+1) subset A_n the
two definitions are equivalent.

> The issue is not whether the naturals are such a limit but whether for
> every so defined limit the cardinality of the limit equals the limit
> cardinality of their cardinalities, which is different sort of limit.


No, actually the issue is whether all possible definitions of N include
a limit, and that is false:
N is the smallest inductive set
involves no limit at all.

And WM's definitions of N did *not* include a limit. He thought so
because he mistakenly thought that an infinite union does imply a
limit. This from the mistaken thought that uniting a collection of
sets goes stepwise.

(And note, Virgil, WM uses the also common notation:
union(i in I) {S_i}
as shorthand for
union{ S_i | i in I}
where the latter is standard ZF. I.e. the uniting is about sets rather
than about members of a single set.
--
dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/



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
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george
12/1/09
Read Re: Another AC anomaly?
george
11/25/09
Read Re: Another AC anomaly?
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
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
Read Re: Another AC anomaly?
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
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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
Read Re: Another AC anomaly?
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
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/26/09
Read Re: Another AC anomaly?
LauLuna
11/26/09
Read Re: Another AC anomaly?
William Hughes
11/26/09
Read Re: Another AC anomaly?
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
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?
Alan Smaill
11/29/09
Read Re: Another AC anomaly?
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
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
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
Read Re: Another AC anomaly?
Dik T. Winter
11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/30/09
Read Re: Another AC anomaly?
Dik T. Winter
11/30/09
Read Re: Another AC anomaly?
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?
William Hughes
12/1/09
Read Re: Another AC anomaly?
Virgil
12/1/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/1/09
Read Re: Another AC anomaly?
Virgil
12/1/09
Read Re: Another AC anomaly?
ross.finlayson@gmail.com
12/1/09
Read Re: Another AC anomaly?
Virgil
12/1/09
Read Re: Another AC anomaly?
ross.finlayson@gmail.com
12/1/09
Read Re: Another AC anomaly?
Virgil
12/2/09
Read Re: Another AC anomaly?
Dik T. Winter
12/2/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/2/09
Read Re: Another AC anomaly?
Dik T. Winter
12/2/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/2/09
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Virgil
12/3/09
Read Re: Another AC anomaly?
Dik T. Winter
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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K_h
12/7/09
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Dik T. Winter
12/7/09
Read Re: Another AC anomaly?
Virgil
12/8/09
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K_h
12/8/09
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Virgil
12/9/09
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K_h
12/9/09
Read Re: Another AC anomaly?
Virgil
12/9/09
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mueckenh@rz.fh-augsburg.de
12/9/09
Read Re: Another AC anomaly?
Virgil
12/9/09
Read Re: Another AC anomaly?
K_h
12/10/09
Read Re: Another AC anomaly?
Dik T. Winter
12/9/09
Read Re: Another AC anomaly?
K_h
12/9/09
Read Re: Another AC anomaly?
Virgil
12/10/09
Read Re: Another AC anomaly?
Dik T. Winter
12/10/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/10/09
Read Re: Another AC anomaly?
Virgil
12/10/09
Read Re: Another AC anomaly?
Dik T. Winter
12/11/09
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mueckenh@rz.fh-augsburg.de
12/11/09
Read Re: Another AC anomaly?
Virgil
12/15/09
Read Re: Another AC anomaly?
Dik T. Winter
12/15/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/15/09
Read Re: Another AC anomaly?
Dik T. Winter
12/15/09
Read Re: Another AC anomaly?
K_h
12/16/09
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Virgil
12/16/09
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mueckenh@rz.fh-augsburg.de
12/16/09
Read Re: Another AC anomaly?
Virgil
12/17/09
Read Re: Another AC anomaly?
Dik T. Winter
12/17/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/17/09
Read Re: Another AC anomaly?
Virgil
12/17/09
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mueckenh@rz.fh-augsburg.de
12/17/09
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YBM
12/17/09
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mueckenh@rz.fh-augsburg.de
12/17/09
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Quaestor
12/18/09
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mueckenh@rz.fh-augsburg.de
12/18/09
Read Re: Another AC anomaly?
Virgil
12/18/09
Read Re: Another AC anomaly?
Dik T. Winter
12/19/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/19/09
Read Re: Another AC anomaly?
Virgil
12/22/09
Read Re: Another AC anomaly?
Dik T. Winter
12/22/09
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Virgil
12/18/09
Read Re: Another AC anomaly?
Dik T. Winter
12/19/09
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mueckenh@rz.fh-augsburg.de
12/19/09
Read Re: Another AC anomaly?
Quaestor
12/21/09
Read Re: Another AC anomaly?
Dik T. Winter
12/21/09
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mueckenh@rz.fh-augsburg.de
12/21/09
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Marshall
12/21/09
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Virgil
12/22/09
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Dik T. Winter
12/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/27/09
Read Re: Another AC anomaly?
Virgil
12/16/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/16/09
Read Re: Another AC anomaly?
Virgil
12/11/09
Read Re: Another AC anomaly?
K_h
12/11/09
Read Re: Another AC anomaly?
Dik T. Winter
12/11/09
Read Re: Another AC anomaly?
K_h
12/11/09
Read Re: Another AC anomaly?
Marshall
12/12/09
Read Re: Another AC anomaly?
Jesse F. Hughes
12/12/09
Read Re: Another AC anomaly?
K_h
12/12/09
Read Re: Another AC anomaly?
K_h
12/11/09
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mueckenh@rz.fh-augsburg.de
12/11/09
Read Re: Another AC anomaly?
Virgil
12/10/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/10/09
Read Re: Another AC anomaly?
Virgil
12/10/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/10/09
Read Re: Another AC anomaly?
Virgil
12/10/09
Read Re: Another AC anomaly?
Dik T. Winter
12/10/09
Read Re: Another AC anomaly?
Dik T. Winter
12/11/09
Read Re: Another AC anomaly?
K_h
12/11/09
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Virgil
12/7/09
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mueckenh@rz.fh-augsburg.de
12/7/09
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Virgil
12/8/09
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Dik T. Winter
12/8/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/8/09
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anonymous.rubbertube@yahoo.com
12/8/09
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Virgil
12/8/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/8/09
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Virgil
12/10/09
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Dik T. Winter
12/3/09
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Virgil
12/3/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/3/09
Read Re: Another AC anomaly?
Virgil
12/7/09
Read Re: Another AC anomaly?
Dik T. Winter
12/7/09
Read Re: Another AC anomaly?
Virgil
12/7/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/7/09
Read Re: Another AC anomaly?
Virgil
12/8/09
Read Re: Another AC anomaly?
Dik T. Winter
12/8/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/8/09
Read Re: Another AC anomaly?
Virgil
12/10/09
Read Re: Another AC anomaly?
Dik T. Winter
12/10/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/10/09
Read Re: Another AC anomaly?
Virgil
12/10/09
Read Re: Another AC anomaly?
Dik T. Winter
12/11/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/11/09
Read Re: Another AC anomaly?
Virgil
12/11/09
Read Re: Another AC anomaly?
Virgil
12/15/09
Read Re: Another AC anomaly?
Dik T. Winter
12/16/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/16/09
Read Re: Another AC anomaly?
Dik T. Winter
12/16/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/17/09
Read Re: Another AC anomaly?
T.H. Ray
12/17/09
Read Re: Another AC anomaly?
Dik T. Winter
12/4/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/4/09
Read Re: Another AC anomaly?
Virgil
12/4/09
Read Re: Another AC anomaly?
Marshall
12/7/09
Read Re: Another AC anomaly?
Dik T. Winter
12/7/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/7/09
Read Re: Another AC anomaly?
Virgil
12/8/09
Read Re: Another AC anomaly?
Dik T. Winter
12/8/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/8/09
Read Re: Another AC anomaly?
Virgil
12/10/09
Read Re: Another AC anomaly?
Dik T. Winter
12/10/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/10/09
Read Re: Another AC anomaly?
Virgil
12/10/09
Read Re: Another AC anomaly?
Dik T. Winter
12/11/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/11/09
Read Re: Another AC anomaly?
Marshall
12/11/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/11/09
Read Re: Another AC anomaly?
Virgil
12/11/09
Read Re: Another AC anomaly?
Virgil
12/11/09
Read Re: Another AC anomaly?
Marshall
12/12/09
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mueckenh@rz.fh-augsburg.de
12/12/09
Read Re: Another AC anomaly?
Virgil
12/12/09
Read Re: Another AC anomaly?
Marshall
12/12/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/12/09
Read Re: Another AC anomaly?
Virgil
12/12/09
Read Re: Another AC anomaly?
Marshall
12/12/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/12/09
Read Re: Another AC anomaly?
Virgil
12/12/09
Read Re: Another AC anomaly?
Marshall
12/12/09
Read Re: Another AC anomaly?
george
12/12/09
Read Re: Another AC anomaly?
Virgil
12/12/09
Read Re: Another AC anomaly?
george
12/13/09
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mueckenh@rz.fh-augsburg.de
12/13/09
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Virgil
12/15/09
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Dik T. Winter
12/16/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/16/09
Read Re: Another AC anomaly?
Virgil
12/17/09
Read Re: Another AC anomaly?
Dik T. Winter
12/17/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/17/09
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Quaestor
12/17/09
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mueckenh@rz.fh-augsburg.de
12/17/09
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Quaestor
12/18/09
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mueckenh@rz.fh-augsburg.de
12/18/09
Read Re: Another AC anomaly?
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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Virgil
12/19/09
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mueckenh@rz.fh-augsburg.de
12/19/09
Read Re: Another AC anomaly?
Marshall
12/19/09
Read Re: Another AC anomaly?
Virgil
12/19/09
Read Re: Another AC anomaly?
Virgil
12/19/09
Read Re: Another AC anomaly?
ross.finlayson@gmail.com
12/19/09
Read Re: Another AC anomaly?
Virgil
12/22/09
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ross.finlayson@gmail.com
12/22/09
Read Re: Another AC anomaly?
Marshall
12/27/09
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ross.finlayson@gmail.com
12/21/09
Read Re: Another AC anomaly?
Dik T. Winter
12/21/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/21/09
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Virgil
12/22/09
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Dik T. Winter
12/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/27/09
Read Re: Another AC anomaly?
Marshall
12/27/09
Read Re: Another AC anomaly?
Virgil
12/27/09
Read Re: Another AC anomaly?
Virgil
12/27/09
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Ralf Bader
12/27/09
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Virgil
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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mueckenh@rz.fh-augsburg.de
12/5/09
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Virgil
12/5/09
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mueckenh@rz.fh-augsburg.de
12/6/09
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Virgil
12/5/09
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Carsten Schultz
12/2/09
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Virgil
12/1/09
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george
12/1/09
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Virgil
11/27/09
Read Re: Another AC anomaly?
Virgil
11/27/09
Read Re: Another AC anomaly?
anonymous.rubbertube@yahoo.com
11/30/09
Read Re: Another AC anomaly?
William Hughes
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
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
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William Hughes
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
Read Re: Another AC anomaly?
William Hughes
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
Virgil

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