Another AC anomaly?
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From: K_h on 11 Dec 2009 02:03 "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote in message news:KuFy3L.Cxt(a)cwi.nl... > In article <QP-dnV0EIYPt2b3WnZ2dnUVZ_h2dnZ2d(a)giganews.com> > "K_h" <KHolmes(a)SX729.com> writes: > > "Virgil" <Virgil(a)home.esc> wrote in message > > news:Virgil-1E8B09.00355309122009(a)newsfarm.iad.highwinds-media.com... > ... > > > The issue between Dik and WM is whether the limit of a > > > sequence of sets > > > according to Dik's definition of such limits is > > > necessarily the same as > > > the limit of the sequence of cardinalities for those > > > sets. > > > > > > And Dik quire successfully gave an example in which > > > the > > > limits differ. > > > > I suspect those definitions are not valid. The > > definition I > > used is the one on wikipedia and is generally > > `standard' -- > > as I've seen it in numerous places, including books and > > websites. > > Have a look at <http://en.wikipedia.org/wiki/Lim_inf> in > the section > titled "Special case: dicrete metric". An example is > given with the > sequence {0}, {1}, {0}, {1}, ... > where lim sup is {0, 1} and lim inf is {}. > > Moreover, in what way can a definition be invalid? It depends on the context. When it comes to supertasks, limsup={0,1} is basically useless. That is why those definitions are not good, and invalid, for evaluating supertasks -- in response to WM's supertask issues. Those definitions are useful only to the extent that they show the limit does not exist: i.e. limsup=/=liminf but that is about all. k
From: K_h on 11 Dec 2009 03:23 "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote in message news:KuFvyG.8r4(a)cwi.nl... > In article <Hv2dnXQ7LtSxUIPWnZ2dnUVZ_hSdnZ2d(a)giganews.com> > "K_h" <KHolmes(a)SX729.com> writes: > > "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote in message > > news:KuAGqH.FrI(a)cwi.nl... > ... > > > Not at all. When you define N as an infinite union > > > there > > > is no limit > > > involved, there is even no sequence involved. N > > > follows > > > immediately > > > from the axioms. > > > > I disagree. Please note that I am not endorsing many of > > WM's claims. There are many equivalent ways of defining > > N. > > I have seen the definition that Rucker uses, in his > > infinity > > and mind book, in a number of books on mathematics and > > set > > theory: On page 240 of his book he defines: > > > > a_(n+1) = a_n Union {a_n} > > > > and then: > > > > a = limit a_n. > > But here an infinite union is *not* involved, that is the > crucial > difference. As stated, you may define N as a limit or > not, and > when it is defined as an infinite union as in: > N = union {1, 2, ..., n} > a limit is not involved. I'm not sure we're on the same page here. The limit set, a, does involve an infinite number of unions; this follows from a_(n+1)=a_nU{a_n}. But an infinite number of unions is also involved in the way I defined N as a limit in my previous post. So I guess I'm unclear what you mean when you write that N is defined as an infinite union by N=union{1,2,...,n}. k
From: Virgil on 11 Dec 2009 04:14 In article <6cCdncLLYcT2nL_WnZ2dnUVZ_qmdnZ2d(a)giganews.com>, "K_h" <KHolmes(a)SX729.com> wrote: > "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote in message > news:KuFvyG.8r4(a)cwi.nl... > > In article <Hv2dnXQ7LtSxUIPWnZ2dnUVZ_hSdnZ2d(a)giganews.com> > > "K_h" <KHolmes(a)SX729.com> writes: > > > "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote in message > > > news:KuAGqH.FrI(a)cwi.nl... > > ... > > > > Not at all. When you define N as an infinite union > > > > there > > > > is no limit > > > > involved, there is even no sequence involved. N > > > > follows > > > > immediately > > > > from the axioms. > > > > > > I disagree. Please note that I am not endorsing many of > > > WM's claims. There are many equivalent ways of defining > > > N. > > > I have seen the definition that Rucker uses, in his > > > infinity > > > and mind book, in a number of books on mathematics and > > > set > > > theory: On page 240 of his book he defines: > > > > > > a_(n+1) = a_n Union {a_n} > > > > > > and then: > > > > > > a = limit a_n. > > > > But here an infinite union is *not* involved, that is the > > crucial > > difference. As stated, you may define N as a limit or > > not, and > > when it is defined as an infinite union as in: > > N = union {1, 2, ..., n} > > a limit is not involved. > > I'm not sure we're on the same page here. The limit set, a, > does involve an infinite number of unions; this follows from > a_(n+1)=a_nU{a_n}. But an infinite number of unions is also > involved in the way I defined N as a limit in my previous > post. So I guess I'm unclear what you mean when you write > that N is defined as an infinite union by > N=union{1,2,...,n}. In ZF, no union can be shown to exist unless there is already a set of all the sets being unioned. Thus where 0 = {} and n+1 = {n,{n}}, one cannot form the union of all naturals unless it is already known (by the axiom of infinity) to exist.
From: Dik T. Winter on 11 Dec 2009 06:57 In article <5ZedndTmvoBac7zWnZ2dnUVZ_qOdnZ2d(a)giganews.com> "K_h" <KHolmes(a)SX729.com> writes: > "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote in message > news:KuFy3L.Cxt(a)cwi.nl... .... > > Have a look at <http://en.wikipedia.org/wiki/Lim_inf> in > > the section > > titled "Special case: dicrete metric". An example is > > given with the > > sequence {0}, {1}, {0}, {1}, ... > > where lim sup is {0, 1} and lim inf is {}. > > > > Moreover, in what way can a definition be invalid? > > It depends on the context. When it comes to supertasks, > limsup={0,1} is basically useless. That is why those > definitions are not good, and invalid, for evaluating > supertasks -- in response to WM's supertask issues. I am not discussing supertasks, nor is WM here. The question is simply whether it is possible that: lim | S_n | != | lim S_n | with some form of limit. And by the definitions I gave (and which you also will find on the wikipedia page above): lim(n -> oo) {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: WM on 11 Dec 2009 09:58
On 11 Dez., 03:28, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <89fb6e91-b6b1-4926-afca-820492e3c...(a)r24g2000yqd.googlegroups..com> WM <mueck...(a)rz.fh-augsburg.de> writes: > > On 10 Dez., 15:40, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > > > Have a look at <http://en.wikipedia.org/wiki/Lim_inf> in the section > > > titled "Special case: dicrete metric". =A0An example is given with the > > > sequence {0}, {1}, {0}, {1}, ... > > > where lim sup is {0, 1} and lim inf is {}. > > > > > > Moreover, in what way can a definition be invalid? > > > > It can be nonsense like the definition: Let N be the set of all > > natural numbers. > > In what way is it nonsense? Either that set does exist or it does not exist. > If it does exist there is indeed such a set, if it does not exist there is > no set satisfying the definition. In both cases the definition is not > nonsense in itself. It is nonsense to define a pink unicorn. The set N does not exist as the union of its finite initial segments. This is shown by the (not existing) path 0.000... in the binary tree. Let {1} U {1, 2} U {1, 2, 3} U ... = {1, 2, 3, ...}. What then is {1} U {1, 2} U {1, 2, 3} U ... U {1, 2, 3, ...} ? If it is the same, then wie have a stop in transfinite counting. If it is not the same, what is it? > > But apparently you are of the opinion that you are only allowed to define > things that do exist. Most essential things in mathematics exist without definitions and, above all, without axioms. Regards, WM |