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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 14, 2009 10:16 AM

In article <k9mdnWn11pFIAbjWnZ2dnUVZ_uWdnZ2d@giganews.com> "K_h" <KHolmes@SX729.com> writes:
> "Jesse F. Hughes" <jesse@phiwumbda.org> wrote in message
> news:878wd7lczh.fsf@phiwumbda.org...

...
> > And that's absolutely correct, as we see above.
>
> Only if the sequences were of the non-naturals {n} not
> sequences of the naturals n.

Eh? The definitions I gave (and which you can find at the wikipedia page
I referred to was about the limit of a sequence of sets.

> > You have some very odd notions yourself. It's a simple
> > application of
> > a perfectly sensible definition of limit.

>
> It violates the spirit of what a limit is in some cases.
> So, although it is sensible, it is not perfectly sensible.

Oh. So give us a definition of limit such that
lim(n->oo) {n} = N
that is sensible (note: a limit of *sets*).

> >> The basic idea of what a limit is suggests that an
> >> appropriate definition for lim(n-->oo){n} should yield
> >> lim(n-->oo){n}={N}:
> >>
> >> {}, {{0}}, {{0,1}}, {{0,1,2}}, {{0,1,2,3}}, ... -->
> >> {{0,1,2,3,4,...}}

Why? (And the first should be {{}}.) But by what definition would that
be valid? (And I ask about the limit of a sequence of *sets*.) What you
are confusing is the limit of a setquence of sets and the limit of the
elements of a sequence of sets.

> The sensibility of a definition is the real issue. Applying
> the so-called standard definitions to {n} leads to a
> cockamamie limit which is at odds with the general notion of
> a limit.

It is not.

> For a better definition, first choose one of the
> wikipedia definitions.

lim sup of the sequence S_0, S_1, ... consists of those elements that
are element of infinitely many S_k.
lim inf of the sequence S_0, S_1, ... consiste of those elements that
are element of all S_k after some k0.
lim exists if lim inf equals lim sup.

> If a sequence of sets, A_n, cannot
> be expressed as {X_n}, for some sequence of sets X_n, then
> lim(n-->oo)A_n is defined by the wikipedia limit.

This makes no sense to me.

> Otherwise
> let L=lim(n-->oo)X_n be the specified wikipedia limit for
> X_n. If L exists then:

So you wish to use different definitions of limits depending on what
the sequence of sets actually is?

>
> lim(n-->oo)A_n = lim(n-->oo){X_n} = {L}
>
> otherwise lim(n-->oo)A_n = lim(n-->oo){X_n} does not exist.
> Under this definition lim(n-->oo){n}={N} and
> |lim(n-->oo){n}|=lim(n-->oo)|{n}|=1. Even this definition
> can be improved. In the spirit of what a good definition of
> a limit should be, we should require that, for example,
> lim(n-->oo){n,n,n}={N,N,N}.

Eh? This is not a limit of sets but a limit of multisets.
--
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