Another AC anomaly?
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From: Dik T. Winter on 10 Dec 2009 09:06 In article <Virgil-1E8B09.00355309122009(a)newsfarm.iad.highwinds-media.com> Virgil <Virgil(a)home.esc> writes: > In article <c8idnc6JG6hv34LWnZ2dnUVZ_q-dnZ2d(a)giganews.com>, > "K_h" <KHolmes(a)SX729.com> wrote: .... > > > In ZF, unions are defined only for sets of sets and for > > > such a set of > > > sets S, the union is defined a the set of all elements of > > > elements of S. > > > > Check out: > > http://planetmath.org/encyclopedia/SequenceOfSetsConvergence.html With that definition set inclusion is a requirement, with the definition I gave it is not. > > - Let N be the limit set formed from the initial set {}. > > > > In this case N is a convergent set: > > > > http://planetmath.org/encyclopedia/SequenceOfSetsConvergence.html > > However, in the discussion between Dik and WM, Dik gave SPECIFIC > definition of what HE meant by the limit of a seqeunce of sets which > differed from that in your citation. The difference is that with my definition (which is also quite generally used) sequences of sets can also converge if there is no set inclusion, as in the sequence {n}. If A_n subset A_(n+1) or A_(n+1) subset A_n the two definitions are equivalent. > The issue is not whether the naturals are such a limit but whether for > every so defined limit the cardinality of the limit equals the limit > cardinality of their cardinalities, which is different sort of limit. No, actually the issue is whether all possible definitions of N include a limit, and that is false: N is the smallest inductive set involves no limit at all. And WM's definitions of N did *not* include a limit. He thought so because he mistakenly thought that an infinite union does imply a limit. This from the mistaken thought that uniting a collection of sets goes stepwise. (And note, Virgil, WM uses the also common notation: union(i in I) {S_i} as shorthand for union{ S_i | i in I} where the latter is standard ZF. I.e. the uniting is about sets rather than about members of a single set. -- 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 10 Dec 2009 09:40 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? -- 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 10 Dec 2009 09:45 In article <k_udnSOIvMxf2r3WnZ2dnUVZ_gqdnZ2d(a)giganews.com> "K_h" <KHolmes(a)SX729.com> writes: .... > > Dik: > > Given a sequence of sets S_n then: > > lim sup{n -> oo} S_n contains those elements that occur > > in > > infinitely > > many S_n > > I don't think this is a good definition for the limit of a > sequence of sets S_n. That is not the definition of a limit but of lim sup. Consider the sequence of numbers 0, 1, 0, 1, ..., the lim sup of this sequence is 1, the lim inf is 0 and as those two are not equal, the limit does not exist. > For instance, consider the > alternating sets: > > S_0 = {0, 2, 4, 6, 8,...} > S_1 = {-1, -3, -5, ...} > S_2 = {0, 2, 4, 6, 8,...} > S_3 = {-1, -3, -5, ...} > S_4 = {0, 2, 4, 6, 8,...} > S_5 = {-1, -3, -5, ...} > ... The lim sup is the set {..., -5, -3, -1, 0, 2, 4, 6, ...}, the lim inf is {} and as these two are not equal the limit does not exist. > Like Thompson's lamp alternating between on and > off it reaches no limit in any intuitive sense of what a > limit is. But it has a lim sup and a lim inf. -- 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 10 Dec 2009 09:56 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". 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 can be nonsense like the definition: Let N be the set of all natural numbers. Regards, WM
From: Dik T. Winter on 10 Dec 2009 10:29
In article <69368271-d841-4c3e-9f73-57259312f585(a)g12g2000yqa.googlegroups.com> WM <mueckenh(a)rz.fh-augsburg.de> writes: > On 8 Dez., 15:22, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: .... > > > I said you can take it from the shelf. It is not defined as a limit > > > (if you like so) although amazingly omega is called a limit ordinal. > > > Yes, it is called a limit ordinal because by definition each ordinal that > > has no predecessor is called a limit ordinal (that is the definition of > > the term "limit ordinal"). It has in itself nothing to do with limits. > > No, that is not the reason. The reason is that omega is a limit > without axiom of infinity, and omega is older than that axiom. Without the axiom of infinity omega would not be immediately existing. So apparently there is a definition of omega without the axiom of infinity. Can you state that definition? > > > N is a concept of mathematics. That's enough. > > > > Yes, and it is a concept of mathematics because it is defined within > > mathematics, and it is not defined as a limit. > > It is a concept of mathematics without any being defined. There are no concepts of mathematics without definitions. > > > The infinite union is a limit. > > > > I do not think you have looked at the definition of an infinite union, if > > you had done so you would find that (in your words) such a union is found > > on the shelf and does not involve limits. Try to start doing mathematics > > and rid yourself of the idea that an infinite union is a limit. > > An infinite union *is* not at all. But if it were, it was a limit. It *is* according to one of the axioms of ZF, and as such it is not a limit. > > > Why did you argue that limits of > > > cardinality and sets are different, if there are no limits at all? > > > > 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. > > That means that you are wrong. Where? Why do you think taking a limit and taking cardinality should commute? Should also the limit of te sequence of integral of functions be equal to the integral of the limit of a sequence of fuctions? -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |