Another AC anomaly?
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From: Virgil on 16 Dec 2009 13:40 In article <6a57309a-a136-430c-a718-e38518c658bb(a)q16g2000vbc.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 16 Dez., 03:44, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > > > �> {1} U {1, 2} U {1, 2, 3} U ... = ({1} U {1, 2} U {1, 2, 3} U ...) U {1, > > �> 2, 3, ...} > > That is a matter of taste. > > > > �> There is �no space in the binary tree to contain infinite paths in > > �> addition to the sequeneces of all finite paths. > > �> The sequence 0.0, 0.00, 0.000, ... when being completely constructed, > > �> is already the path 0.000... > > �> You cannot add it separately. > > > > A sequence of paths is not a path. > But a union of paths is. Not always. And the union of infinitely many different paths, if a path at all, is necessarily an infinite path, despite all of WM's counterclaims. > > > > �> Therefore: When all finite paths have been constructed within aleph_0 > > �> steps, then all paths have been constructed. > > > > Here, again, you err. �You can not construct something in aleph_0 steps; you > > will never complete your construction. �You *cannot* get at aleph_0 step by > > step. > > But you can make a bijection with all elements of omega? Yes, because it does not have to be done sequentially but can be done globally. > > > > �> This hold for every limit of every sequence of finite paths. > > > > A limit is not a step by step process. > > Then assume it is a mapping from omega. If one is allowed to assume that, then all of WM's claims go down the tubes along with that assumption. As soon as WM concedes the existence of omega, his claims all wither and die.
From: Virgil on 16 Dec 2009 13:53 In article <671ef81e-0cf8-42e7-a4f5-0c552646cc24(a)f6g2000vbp.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 16 Dez., 04:56, "K_h" <KHol...(a)SX729.com> wrote: > > "WM" <mueck...(a)rz.fh-augsburg.de> wrote in message > > > > news:9904ae9d-de2c-4d5f-86da-be165e8e9d7e(a)p30g2000vbt.googlegroups.com... > > On 15 Dez., 13:32, "Dik T. Winter" <Dik.Win...(a)cwi.nl> > > wrote: > > > > > > > > > Look two lines above: > > > > > Let {1} U {1, 2} U {1, 2, 3} U ...={1, 2, 3, ...} > > > hence > > > {1} U {1, 2} U {1, 2, 3} U ... U {1, 2, 3, ...}={1, 2, 3, > > > ...} . > > > > All limit ordinals, X, satisfy UX=X. �That means that the > > union of the members of a limit ordinal is equal to the > > ordinal itself. �w=N is a limit ordinal so UN=N. �So: > > > > {} U {0} U {0,1,} U {0,1 2,} U ... = {0,1,2,3, ...} > > > > is a true equation. > > Therefore 0.000..., as a path in the tree, is nothing but the union of > its finite initial segments. pi is also nothing but the union of its > finite initial segments. > > All finite initial segments form a countable set. Nothing else exists. If nothing else exists, then those unions of finite initial segments which represent other than binary rationals do not exist, and thus more reals do not exist. > Hence all paths in the tree form a countable set. Not if it is a COMPLETE infinite binary tree, which necessarily contains infinite strings not in WM's trees. Consider this tree: Each n in the set of all naturals , N, is a node. For each such node, n, in N, its left and right children are, respectively, 2*n and 2*n+1. A path, P, is, by definition, a subset of N such that 1 is a member of P, and, for each n in P, exactly one of 2*n and 2*n+1 is a member of P. The set of all such paths is easily shown, by a version of the Cantor diagonal argument, not to allow bijection with N. Thus the set of all such paths is demonstrably uncountable.
From: K_h on 16 Dec 2009 19:44 "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote in message news:KuqwG3.qv(a)cwi.nl... > In article <k6idnaWIQNah0LXWnZ2dnUVZ_hqdnZ2d(a)giganews.com> > "K_h" <KHolmes(a)SX729.com> writes: > > > > "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote in message > > news:Kup2G0.IBK(a)cwi.nl... > > > In article > > > <iNWdnfmPh7NpQ7vWnZ2dnUVZ_hydnZ2d(a)giganews.com> > > > "K_h" <KHolmes(a)SX729.com> writes: > > > > "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote in message > > > > news:KunEFz.913(a)cwi.nl... > > > ... > > > > > > The definition you provided for a sequence of sets A_n > > > depends on whether > > > each A_n is or is not a set containing a single set as > > > an > > > element. > > > > > > Your definition leads to some strange consequences. I > > > can > > > state the > > > following theorem: > > > > > > Let A_n and B_n be two sequences of sets. Let A_s = > > > lim > > > sup A_n and > > > A_i = lim inf A_n, similar for B_s and B_i. Let C_n > > > be > > > the sequence > > > defined as: > > > C_2n = A_n > > > C_(2n+1) = B_n > > > Theorem: > > > lim sup C_n = union (A_s, B_s) > > > lim inf C_n = intersect (A_i, B_i) > > > Proof: > > > easy. > > > > Yes, my definition did not include a limsup and liminf > > but > > they can be added. With this addition, the limit of > > sets > > like {X_n} is more in line with the general notion of a > > limit. > > Well, the above theorem is still not valid with your > definition. What case did you have in mind? I found cases where it works fine. k
From: K_h on 17 Dec 2009 01:20 "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote in message news:KuqwqJ.24o(a)cwi.nl... > In article <hN-dneOj6K8oz7XWnZ2dnUVZ_rKdnZ2d(a)giganews.com> > "K_h" <KHolmes(a)SX729.com> writes: > > "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote in message > > news:Kuq5DH.18H(a)cwi.nl... > ... > > The general idea of a limit is that the limiting state > > is > > what you get when you go through all sequences. If one > > defines the naturals as you have done above then the > > general > > notion of a limit suggests that the limiting state > > should be > > something like: > > > > {...{{{{{{...{}...}}}}}}...} = limit > > > > We could construct a defintion of a limit so that this > > is > > the end result but it may be that a better definition > > for > > the limiting case of 0={} and n+1={n}is a defintion > > where > > lim(n -> oo)n does not exist. > > We are talking about lim(n -> oo) {n} which is the limit > of a sequence of > sets, and not about lim(n -> oo) n which may or may not be > the limit of > a sequence of sets, depending on the actual construction > of the natural > numbers. For the sets n, lim(n-->oo)n is the limit of a sequence of those sets. What lim(n-->oo){n}={} really means is that each {n} does not persist after it first appears. The essence of the defined limsup and liminf is that they only contain those sets that persist, as members of subsequent sets, after they first appear. For the non-naturals, {n}, lim(n-->oo){n}={} just says nothing is accumulated. For the naturals, n, lim(n-->oo)n=N just says that everything is accumulated. There are many ways a limit of a sequence of sets can be defined. Wikipedia gives two such examples but these are not the only two options. Using the above notion of accumulation, lim(n-->oo){A_n} can be defined by a set containing the accumulation of A_n. For example, this definition says lim(n-->oo){n}={N} and its sequence and limit look like. {{}}, {{0}}, {{0,1}}, {{0,1,2}}, ... --> {{0,1,2,3,,...}} So this definition is motivated by the intuitive idea of what a limit is in cases like these. Like many definitions, this one has disadvantages: some theorems true in other definitions will not be true with this one. > > > > Theorem: > > > > lim(n ->oo) n = N. Consider the naturals: > > > > > > > > S_0 = 0 = {} > > > > S_1 = 1 = {0} > > > ... > > > This presupposes a particular construction for the > > > natural > > > number. There are > > > other constructions that are consistent with ZF. Is > > > the > > > limit valid for all > > > those possible models? > > > > > Why do you ask? There are many ways a limit can be > > defined > > in ZF but the definition should embody the general idea > > of > > what a limit is. > > But the definition ought to be such that the limit of a > sequence does not > depend on the exact construction of the sequence. That is > that > lim(n -> oo) {n} > should be independent on the way the natural numbers are > constructed. lim(n ->oo) n = N is true for the standard definition of natural numbers using just the wikipedia definitions for the limit of a sequence of sets. k
From: Dik T. Winter on 17 Dec 2009 06:45
In article <GLidnXhuo5t347TWnZ2dnUVZ_sWdnZ2d(a)giganews.com> "K_h" <KHolmes(a)SX729.com> writes: > "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote in message > news:KuqwG3.qv(a)cwi.nl... .... > > > > The definition you provided for a sequence of sets A_n > > > > depends on whether > > > > each A_n is or is not a set containing a single set as > > > > an > > > > element. > > > > > > > > Your definition leads to some strange consequences. I > > > > can > > > > state the > > > > following theorem: > > > > > > > > Let A_n and B_n be two sequences of sets. Let A_s = > > > > lim > > > > sup A_n and > > > > A_i = lim inf A_n, similar for B_s and B_i. Let C_n > > > > be > > > > the sequence > > > > defined as: > > > > C_2n = A_n > > > > C_(2n+1) = B_n > > > > Theorem: > > > > lim sup C_n = union (A_s, B_s) > > > > lim inf C_n = intersect (A_i, B_i) > > > > Proof: > > > > easy. .... > > Well, the above theorem is still not valid with your > > definition. > > What case did you have in mind? I found cases where it > works fine. Let's have some arbitrary object 'a' and the natural numbers. Create the sequence A_n where A_n = {a} and the sequence B_n where B_n = {n}. According to your definition: lim sup A_n = {a} and lim sup B_n = {N}. Now create the sequence C_n: C_2n = A_n, C_2n+1 = B_n. Again according to your definition: lim sup C_n = {a} which is not equal to union (lim sup A_n, lim sup B_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/ |