
Re: Another AC anomaly?
Posted:
Dec 15, 2009 7:52 AM


In article <iNWdnfmPh7NpQ7vWnZ2dnUVZ_hydnZ2d@giganews.com> "K_h" <KHolmes@SX729.com> writes: > "Dik T. Winter" <Dik.Winter@cwi.nl> wrote in message > news:KunEFz.913@cwi.nl... ... > > > >> 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}}, {{0,1}}, {{0,1,2}}, {{0,1,2,3}}, ...> > > > >> {{0,1,2,3,4,...}} > > > > Why? > > Why not?
See below.
> > > 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. > > Why not?
See below.
> > > 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? > > No, the defintion I provided is one defintion that includes > stuff from the wikipedia definition.
The definition you provided for a sequence of sets A_n depends on whether each A_n is or is not a set containing a single set as an element.
Your definition leads to some strange consequences. I can state the following theorem:
Let A_n and B_n be two sequences of sets. Let A_s = lim sup A_n and A_i = lim inf A_n, similar for B_s and B_i. Let C_n be the sequence defined as: C_2n = A_n C_(2n+1) = B_n Theorem: lim sup C_n = union (A_s, B_s) lim inf C_n = intersect (A_i, B_i) Proof: easy.
However with your definition for a sequence of sets depending on whether the terms of the sequence are or are not a set containing a single set as element, this theorem does not hold.  dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/

