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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/
From: Ilmari Karonen on 14 Dec 2009 16:27 ["Followup-To:" header set to sci.math.] On 2009-12-14, Jesse F. Hughes <jesse(a)phiwumbda.org> wrote: > "Jesse F. Hughes" <jesse(a)phiwumbda.org> writes: >> >> 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? Well, not really. That's not the same as the definition of general set convergence, but I do believe the two definitions are equivalent for sequences of natural numbers, at least under any of the usual set-theoretic constructions of the naturals. In particular, under the standard construction of the naturals, where 0 = {} and n+1 = n union {n}, I believe the two definitions of lim sup and lim inf also match: this is due to the fact that, for the natural numbers m and n under this construction, m is a subset of n if and only if m <= n. -- Ilmari Karonen To reply by e-mail, please replace ".invalid" with ".net" in address.
From: K_h on 14 Dec 2009 19:57
"Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote in message news:KunEFz.913(a)cwi.nl... > 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... > ... > > >> 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? > (And the first should be {{}}.) Yes, my mistake; corrected above. > > 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. Why not? > > > > 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. > > 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. Good point. k |