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Topic: Another AC anomaly?
Replies: 43   Last Post: Dec 21, 2009 8:08 AM

 Messages: [ Previous | Next ]
 Dik T. Winter Posts: 7,899 Registered: 12/6/04
Re: Another AC anomaly?
Posted: Dec 15, 2009 7:52 AM

In article <iNWdnfmPh7NpQ7vWnZ2dnUVZ_hydnZ2d@giganews.com> "K_h" <KHolmes@SX729.com> writes:
> "Dik T. Winter" <Dik.Winter@cwi.nl> wrote in message
> news:KunEFz.913@cwi.nl...

...
> > > >> The basic idea of what a limit is suggests that an
> > > >> appropriate definition for lim(n-->oo){n} should
> > > >> yield
> > > >> lim(n-->oo){n}={N}:
> > > >>
> > > >> {0}, {{0}}, {{0,1}}, {{0,1,2}}, {{0,1,2,3}}, ...-->
> > > >> {{0,1,2,3,4,...}}

> >
> > Why?

>
> Why not?

See below.

> > > The sensibility of a definition is the real issue.
> > > Applying
> > > the so-called standard definitions to {n} leads to a
> > > cockamamie limit which is at odds with the general
> > > notion of
> > > a limit.

> >
> > It is not.

>
> Why not?

See below.

> > > Otherwise
> > > let L=lim(n-->oo)X_n be the specified wikipedia limit
> > > for
> > > X_n. If L exists then:

> >
> > So you wish to use different definitions of limits
> > depending on what
> > the sequence of sets actually is?

>
> No, the defintion I provided is one defintion that includes
> stuff from the wikipedia definition.

The definition you provided for a sequence of sets A_n depends on whether
each A_n is or is not a set containing a single set as an element.

Your definition leads to some strange consequences. I can state the
following theorem:

Let A_n and B_n be two sequences of sets. Let A_s = lim sup A_n and
A_i = lim inf A_n, similar for B_s and B_i. Let C_n be the sequence
defined as:
C_2n = A_n
C_(2n+1) = B_n
Theorem:
lim sup C_n = union (A_s, B_s)
lim inf C_n = intersect (A_i, B_i)
Proof:
easy.

However with your definition for a sequence of sets depending on whether
the terms of the sequence are or are not a set containing a single set
as element, this theorem does not hold.
--
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
12/12/09 Jesse F. Hughes
12/13/09 K_h
12/14/09 Dik T. Winter
12/14/09 K_h
12/15/09 Dik T. Winter
12/15/09 K_h
12/16/09 Dik T. Winter
12/16/09 K_h
12/17/09 Dik T. Winter
12/18/09 K_h
12/18/09 Dik T. Winter
12/18/09 K_h
12/21/09 Dik T. Winter
12/14/09 Dik T. Winter
12/14/09 K_h
12/15/09 Dik T. Winter
12/15/09 Dik T. Winter
12/15/09 K_h
12/16/09 Dik T. Winter
12/17/09 K_h
12/17/09 Dik T. Winter
12/18/09 K_h
12/18/09 K_h
12/18/09 Dik T. Winter
12/18/09 K_h
12/18/09 Dik T. Winter
12/18/09 K_h
12/15/09 K_h
12/16/09 Jesse F. Hughes
12/17/09 Dik T. Winter
12/17/09 Jesse F. Hughes
12/16/09 Dik T. Winter
12/15/09 ross.finlayson@gmail.com
12/13/09 K_h
12/13/09 Jesse F. Hughes
12/13/09 Jesse F. Hughes
12/14/09 Ilmari Karonen
12/14/09 Jesse F. Hughes
12/15/09 Chas Brown