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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 16, 2009 7:44 AM

In article <hN-dneOj6K8oz7XWnZ2dnUVZ_rKdnZ2d@giganews.com> "K_h" <KHolmes@SX729.com> writes:
> "Dik T. Winter" <Dik.Winter@cwi.nl> wrote in message
> news:Kuq5DH.18H@cwi.nl...

...
> > > 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.
> >
> > Even with your definition
> > 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}={}?

That is not important for me. With *your* definition and the construction
of the natural numbers with n+1 = {n}, we *get* that lim(n -> oo) {n} = {}
(note: set limit).

> 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.

We are talking about lim(n -> oo) {n} which is the limit of a sequence of
sets, and not about lim(n -> oo) n which may or may not be the limit of
a sequence of sets, depending on the actual construction of the natural
numbers.

But if I understand you well, your opinion is not that lim(n -> oo) {n}
can be {N} or non-existing, depending on the way the natural numbers
are constructed?
--
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