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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 15, 2009 10:23 PM

"Dik T. Winter" <Dik.Winter@cwi.nl> wrote in message
news:Kuq5DH.18H@cwi.nl...
> In article <Kup2uH.JC7@cwi.nl> "Dik T. Winter"
> <Dik.Winter@cwi.nl> writes:

> > In article
> > <r_6dnUXDv4pMfbvWnZ2dnUVZ_tWdnZ2d@giganews.com> "K_h"
> > <KHolmes@SX729.com> writes:

> > > "Dik T. Winter" <Dik.Winter@cwi.nl> wrote in message
> > > news:KunF0v.9y9@cwi.nl...

> > ...
> > > > By what definition is:
> > > > lim(n -> oo) n = N?

> > >
> > > Any of the wikipedia definitions. Here is the proof

> > again.
> > ...

> > > Theorem:
> > > lim(n ->oo) n = N. Consider the naturals:
> > >
> > > S_0 = 0 = {}
> > > S_1 = 1 = {0}

> > ...
> > This presupposes a particular construction for the
> > natural number. There are
> > other constructions that are consistent with ZF. Is the
> > limit valid for all
> > those possible models?

>
> For starters, try it with
> 0 = {}
> n+1 = {n}
> which is a valid construction of the naturals in ZF.
>
> lim sup(n -> oo) {n} = {}

Why is it so important to you to have a limit definition and
a construction of the naturals such that lim(n->oo){n}={}?
The general idea of a limit is that the limiting state is
what you get when you go through all sequences. If one
defines the naturals as you have done above then the general
notion of a limit suggests that the limiting state should be
something like:

{...{{{{{{...{}...}}}}}}...} = limit

We could construct a defintion of a limit so that this is
the end result but it may be that a better definition for
the limiting case of 0={} and n+1={n}is a defintion where
lim(n -> oo)n does not exist.

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