
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 nonnaturals {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 socalled 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/

