Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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 17, 2009 1:20 AM

"Dik T. Winter" <Dik.Winter@cwi.nl> wrote in message
news:KuqwqJ.24o@cwi.nl...
> 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...

> ...
> > 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.

For the sets n, lim(n-->oo)n is the limit of a sequence of
those sets. What lim(n-->oo){n}={} really means is that
each {n} does not persist after it first appears. The
essence of the defined limsup and liminf is that they only
contain those sets that persist, as members of subsequent
sets, after they first appear. For the non-naturals, {n},
lim(n-->oo){n}={} just says nothing is accumulated. For the
naturals, n, lim(n-->oo)n=N just says that everything is
accumulated. There are many ways a limit of a sequence of
sets can be defined. Wikipedia gives two such examples but
these are not the only two options. Using the above notion
of accumulation, lim(n-->oo){A_n} can be defined by a set
containing the accumulation of A_n. For example, this
definition says lim(n-->oo){n}={N} and its sequence and
limit look like.

{{}}, {{0}}, {{0,1}}, {{0,1,2}}, ... --> {{0,1,2,3,,...}}

So this definition is motivated by the intuitive idea of
what a limit is in cases like these. Like many definitions,
this one has disadvantages: some theorems true in other
definitions will not be true with this one.

> > > > 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?
> > >

> > Why do you ask? There are many ways a limit can be
> > defined
> > in ZF but the definition should embody the general idea
> > of
> > what a limit is.

>
> But the definition ought to be such that the limit of a
> sequence does not
> depend on the exact construction of the sequence. That is
> that
> lim(n -> oo) {n}
> should be independent on the way the natural numbers are
> constructed.

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.

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