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

 Messages: [ Previous | Next ]
 K_h Posts: 419 Registered: 4/12/07
Re: Another AC anomaly?
Posted: Dec 18, 2009 3:26 AM

"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 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