K_h
Posts:
419
Registered:
4/12/07


Re: Another AC anomaly?
Posted:
Dec 13, 2009 9:02 PM


"Jesse F. Hughes" <jesse@phiwumbda.org> wrote in message news:878wd7lczh.fsf@phiwumbda.org... > "K_h" <KHolmes@SX729.com> writes: > >> "Jesse F. Hughes" <jesse@phiwumbda.org> wrote in message >> news:87vdgcksy2.fsf@phiwumbda.org... >>> "K_h" <KHolmes@SX729.com> writes: >>> >> >> So, you're claiming that he is not using { and } just to >> bracket the argument (i.e. the X_n to be limited) but >> {X_n} >> refers to a set containing the one set X_n. > > Er, yes. Though, something seems to be wrong with your > notation. At > issue is the set X_n = {n}, not the set {X_n}. > > In summary: > > lim X_n = lim {n} = {} = 0. > > lim X_n = lim {n} = lim 1 = 1. > >> Then it seems like the meaning has changed because in >> previous posts >> he writes that limS_n=/=limS_n follows from a >> wikipedia >> definition applied to sequences of natural numbers n  >> not to the >> nonnaturals {n}. For example: >> >> > > I have explicitly defined the limit of a sequence of >> sets. With that >> > > definition (and the common definition of limits of >> sequences of natural >> > > numbers) I found that the cardinality of the limit >> is >> not necessarily >> > > equal to the limit of the cardinalities. > > And that's absolutely correct, as we see above.
Only if the sequences were of the nonnaturals {n} not sequences of the naturals n.
>> Okay, if {X_n} refers to a set containing the single set >> X_n >> then lim(n>oo){n} is not a limit of the natural numbers >> since the naturals are not the sets {n} but the sets n. > > Er, yes. Of course. > >> In this case my proof shows that lim(n>oo)n=N. >> Applying the >> wikipedia definitions to n is sensible but applying them >> to {n} >> makes a mockery of the notion of a limit. > > 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.
>> The basic idea behind a limit is that things in one state >> tend to >> some final state and a good definition and application of >> a limit >> should embody that. In looking at the sequence {n}, with >> 0={} and >> 1={0}, saying that it tends to 0={} is a betrayal of the >> core idea >> behind a limit: >> >> 1, {{0}}, {{0,1}}, {{0,1,2}}, {{0,1,2,3}}, ... > 0 >> >> 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,...}} >> >> In other words, applying the wikipedia definitions to {n} >> is >> an abuse of those definitions. The definition that is >> used >> for a limit should make sense for the kind of object it >> is >> applied to. > > You're welcome to your own cockamamie opinions about > whether a > particular definition is sensible or not, but they're > utterly > irrelevant to the issue at hand. The fact is that with > this > *perfectly standard* definition of limits, we see that > > lim X_n != lim X_n. > > That's all there was at issue.
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. For a better definition, first choose one of the wikipedia definitions. 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. Otherwise let L=lim(n>oo)X_n be the specified wikipedia limit for X_n. If L exists then:
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}. This can be done by a simple generalization: if a sequence of sets A_n cannot be expressed as {X_n,Y_n,Z_n,...}, for one or more arguments, then lim(n>oo)A_n is defined by the specified wikipedia limit. Otherwise let L=lim(n>oo)X_n, K=lim(n>oo)Y_n, J=lim(n>oo)Z_n, ... be the specified wikipedia limits for X_n, Y_n, Z_n, ... . If L, K, J, .. all exist then:
lim(n>oo)A_n = lim(n>oo){X_n,Y_n,Z_n,...} = {L,K,J,...}
otherwise lim(n>oo)A_n does not exist. Viewers of this thread may want to see if there is a way to generalize and/or improve this definition further. A good definition should always seek to capture the essence of the notion it is defining.
k

