Another AC anomaly?
in [Logic]
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From: K_h on 13 Dec 2009 21:03 "Jesse F. Hughes" <jesse(a)phiwumbda.org> wrote in message news:878wd7lczh.fsf(a)phiwumbda.org... > "K_h" <KHolmes(a)SX729.com> writes: > >> "Jesse F. Hughes" <jesse(a)phiwumbda.org> wrote in message >> news:87vdgcksy2.fsf(a)phiwumbda.org... >>> "K_h" <KHolmes(a)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 lim|S_n|=/=|limS_n| follows from a >> wikipedia >> definition applied to sequences of natural numbers n -- >> not to the >> non-naturals {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 non-naturals {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 so-called 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
From: Jesse F. Hughes on 13 Dec 2009 22:31 "K_h" <KHolmes(a)SX729.com> writes: > It violates the spirit of what a limit is in some cases. So, > although it is sensible, it is not perfectly sensible. Yeah, the discrete topology is different than other topologies. But the standard topology on N is the discrete topology, too! Thus, the standard definition of sequence convergence on N is inherited via the subspace topology from Set. That is, a sequence {a_n | n in N} c N converges (in N) to m iff (E k)(A j > k) a_j = m. This is (unless I'm just butt-wrong) the same as the definition of sequence convergence on Set restricted to the subspace N. So it seems to me that you either dislike the discrete topology (which is certainly a bit odd) or you just dislike the discrete topology on Set. The latter opinion is a bit odd, too. -- "[Criticizing JSH's mathematics will result in] one of the worst debacles in the history of the world. It is foretold in most mythologies and religions. And yes, you are the ones, the cursed ones, who destroy the world." --James S. Harris reads from the Aztec Book of the Damned Mathematicians
From: Jesse F. Hughes on 13 Dec 2009 22:52 "Jesse F. Hughes" <jesse(a)phiwumbda.org> writes: > "K_h" <KHolmes(a)SX729.com> writes: > >> It violates the spirit of what a limit is in some cases. So, >> although it is sensible, it is not perfectly sensible. > > Yeah, the discrete topology is different than other topologies. > > But the standard topology on N is the discrete topology, too! Thus, > the standard definition of sequence convergence on N is inherited via > the subspace topology from Set. That is, a sequence > {a_n | n in N} c N converges (in N) to m iff > > (E k)(A j > k) a_j = m. > > This is (unless I'm just butt-wrong) the same as the definition of > sequence convergence on Set restricted to the subspace N. Yeah, well, I am just butt-wrong, ain't I? I was confusing the discrete space of Set with what the Wikipedia article calls "general set convergence". My mistake. -- Jesse F. Hughes "You know that view most people have of mathematicians as brilliant people? What if they're not?" -- James S. Harris
From: Dik T. Winter on 14 Dec 2009 10:16 In article <k9mdnWn11pFIAbjWnZ2dnUVZ_uWdnZ2d(a)giganews.com> "K_h" <KHolmes(a)SX729.com> writes: > "Jesse F. Hughes" <jesse(a)phiwumbda.org> wrote in message > news:878wd7lczh.fsf(a)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/
From: Dik T. Winter on 14 Dec 2009 10:29
In article <k9mdnWn11pFIAbjWnZ2dnUVZ_uWdnZ2d(a)giganews.com> "K_h" <KHolmes(a)SX729.com> writes: Let's see whether I do understand what you want: .... > 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. That is your opinion. > For a better definition, first choose one of the > wikipedia definitions. We better choose a definition that fits, let's take the definition for sets with discrete metric on the elements. > 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. So with that definition lim sup(n -> oo) {1/n} = {} (note: we use a discrete metric on the rational numbers). > 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} By what definition is it {N}? By what definition is: lim(n -> oo) n = N? -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |