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

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K_h

Posts: 419
Registered: 4/12/07
Re: Another AC anomaly?
Posted: Dec 18, 2009 3:26 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply


"Dik T. Winter" <Dik.Winter@cwi.nl> wrote in message
news:Kusoty.176@cwi.nl...
> In article <yrydnX_FwfczULTWnZ2dnUVZ_vmdnZ2d@giganews.com>
> "K_h" <KHolmes@SX729.com> writes:
> ...

> > lim(n ->oo) n = N is true for the standard definition of
> > natural numbers using just the wikipedia definitions for
> > the
> > limit of a sequence of sets.

>
> But not with Zermelo's definition of the natural numbers.
> Nor when we take
> the natural numbers as embedded in the rational numbers.


Yes, and that was never denied. I just point out that any
limit definition for a sequence of sets will give answers
that violate the intuitive notion of a limit for certain
constructions. There are two ways one can deal with that.
First, for a given class of constructions, have another
definition that does not violate the limit notion. The
second way is to look at the meaning that the wikipedia
definitions have for the constructions. I now think that
the latter approach is superior to the first since the first
caused so much misunderstanding. With the second option we
can restrict ourselves just to the wikipedia definitions and
define the sets n by:

n = 0 = {}
n = 1 = {{}}
n = 2 = {{{}}}
...

Wikipedia gives liminf(n-->oo) n = 0. What this means is
that nothing is `accumulated' in a limit set since each set
does not persist past its introduction. So the notion of n
growing bigger, as one tends to the limit, is not embodied
here since |n| is 0,1,1,1,1... as one proceeds and is 0 in
the limiting case. Under the standard construction each
natural is `accumulated' in the limit set, since each set
persists beyond its introduction, and this preserves the
notion of n growing bigger as one proceeds: |n| is
0,1,2,3,... and is N in the limiting case.

k





Date Subject Author
12/12/09
Read Re: Another AC anomaly?
Jesse F. Hughes
12/13/09
Read Re: Another AC anomaly?
K_h
12/14/09
Read Re: Another AC anomaly?
Dik T. Winter
12/14/09
Read Re: Another AC anomaly?
K_h
12/15/09
Read Re: Another AC anomaly?
Dik T. Winter
12/15/09
Read Re: Another AC anomaly?
K_h
12/16/09
Read Re: Another AC anomaly?
Dik T. Winter
12/16/09
Read Re: Another AC anomaly?
K_h
12/17/09
Read Re: Another AC anomaly?
Dik T. Winter
12/18/09
Read Re: Another AC anomaly?
K_h
12/18/09
Read Re: Another AC anomaly?
Dik T. Winter
12/18/09
Read Re: Another AC anomaly?
K_h
12/21/09
Read Re: Another AC anomaly?
Dik T. Winter
12/14/09
Read Re: Another AC anomaly?
Dik T. Winter
12/14/09
Read Re: Another AC anomaly?
K_h
12/15/09
Read Re: Another AC anomaly?
Dik T. Winter
12/15/09
Read Re: Another AC anomaly?
Dik T. Winter
12/15/09
Read Re: Another AC anomaly?
K_h
12/16/09
Read Re: Another AC anomaly?
Dik T. Winter
12/17/09
Read Re: Another AC anomaly?
K_h
12/17/09
Read Re: Another AC anomaly?
Dik T. Winter
12/18/09
Read Re: Another AC anomaly?
K_h
12/18/09
Read Re: Another AC anomaly?
K_h
12/18/09
Read Re: Another AC anomaly?
Dik T. Winter
12/18/09
Read Re: Another AC anomaly?
K_h
12/18/09
Read Re: Another AC anomaly?
Dik T. Winter
12/18/09
Read Re: Another AC anomaly?
K_h
12/15/09
Read Re: Another AC anomaly?
K_h
12/16/09
Read Re: Another AC anomaly?
Jesse F. Hughes
12/17/09
Read Re: Another AC anomaly?
Dik T. Winter
12/17/09
Read Re: Another AC anomaly?
Jesse F. Hughes
12/16/09
Read Re: Another AC anomaly?
Dik T. Winter
12/15/09
Read Re: Another AC anomaly?
ross.finlayson@gmail.com
12/13/09
Read Re: Another AC anomaly?
K_h
12/13/09
Read Re: Another AC anomaly?
Jesse F. Hughes
12/13/09
Read Re: Another AC anomaly?
Jesse F. Hughes
12/14/09
Read Re: Another AC anomaly?
Ilmari Karonen
12/14/09
Read Re: Another AC anomaly?
Jesse F. Hughes
12/15/09
Read Re: Another AC anomaly?
Chas Brown

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