Cantor Confusion
in [Math]
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From: Dik T. Winter on 9 Mar 2007 08:05 In article <1173430532.626624.124950(a)c51g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > On 8 Mrz., 16:25, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: .... > > Ok, so C(oo) is the cardinal number aleph-0. .... > > > I defined for several times: > > > lim {n-->oo} {1,2,3,...,n} = N > > > lim {n-->oo} |{1,2,3,...,n}| = aleph_0 > > > > > > lim {n-->oo} {2,4,6,...,2n} = set of all even numbers > > > lim {n-->oo} |{2,4,6,...,2n}| = aleph_0 > > > > Ah, you need *four* definitions here? Only two would be sufficient, I > > think. Because none of the definitions follows from any of the other > > definitions. > > > > > The limit lim {n-->oo} n is a number which n comes as close as you > > > like. This is not omega or any greater number. If it exists, then it > > > can only be less. > > > > So, according to *your* definitions it is >. Not according to common > > definitions. Would it not be possible that your definitions are not > > consistent? It is your last paragraph which is inconsistent with the > > other definitions. See how you *did* define: > > > lim {n-->oo} |{1,2,3,...,n}| = aleph_0 > > which means (as |{1,2,3,...,n}| = n: > > lim {n-->oo} n = aleph_0? Do you not have a comment on this? > > Why? I see only that C(oo) is aleph-0, but I see no relation between > > paths and aleph-0. > > Isn't it difficult to look from this perspectice? The cross sections > *are* numbers of paths. No, they are *not*. The cross sections are sets of nodes (by your own definition), except for C(oo), which is a cardinal number (again by your own definition). And I see *no* relation between aleph-0 and the paths. > 1) We know tha every digi of a real number stands on a finite place. > (We know that here is no digit the last one, but that is here > completely irrelevant.) Indeed. > 2) From (1) we obtain that every node of a path is at a finite place. > That means, there is no part of a path which would jut out f the tree. Indeed. > 3) All paths which are in the tree cross each cross section (that is > why it is named cross section). Except for C(oo), because that is not a set of nodes, so there is *no* path that crosses it. > There is no path or even splitting of paths outside of every cross > section. > > Therefore, the limit processes for paths-lengths and cross sections > are identical. How do you *define* that limit process? Your continuous mentioning limits and limit processes show that you do not understand how things work. T(oo) is the union of a collection of trees T(n) (where each tree is a set of nodes). In the *definition* of that union there is no limit involved. Also the infinite paths are unions of paths (again, these are sets of nodes). And again, there is no limit involved in that definition. There is *no* standard definition of limits where the limit of path lengths is aleph-0. You have to provide that definition. There is also no standard definition of limits that allows you to take the limit of cross sections (which would be a set of nodes), so you have to provide that definition. Finally, you have to *prove* that what the limit (by your definition) gives is also the actual value. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 9 Mar 2007 10:07 In article <1173438655.923501.184140(a)8g2000cwh.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > On 7 Mrz., 16:01, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article <1173215257.272668.121...(a)64g2000cwx.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > > > Neither do refer to limits, unless you *define* those limits. > > For the cross section we would not even need a limit. It is enough to > know that at every level n (with finite n) the number of paths (- > bundles) is countable. That is sufficient because we know that every > path consists only of nodes (or edges) at finite places, i.e., > enumerated with finite numbers n. How many of these levels are > populated by paths is completely irrelevant. And this tells us precisely nothing about C(oo), which you *do* use. > > > Have you really been blinded for the fact that most of the subsets of > > > nodes of the complete cannot serve as paths? > > > > That does not matter. Some subsets are paths. Of these some are infinite > > and some are finite. The finite ones are a subset of a countable set, the > > infinite ones are a subset of an uncountable set. But there *do* exist > > uncountable subsets of uncountable sets. > > Again and again the creed is muttered. I thought you were trying to find a contradiction in set theory? Unless you have proven yourself that that is not the case, it will stand. > > > Part of which argument? I cannot see any argument at all. You state > > > (1) that Cantor's argument is valid (at this place I do no judge about > > > it) and you state (2) that set theory is free of contradictions (I > > > don't know who told you). From (1) and (2) you conclude that contrary > > > to all mathematical evidence (I remind you of the ration of 2 nodes > > > per path and the cross section of the tree which counts its paths) my > > > result cannot be true. > > > > As you do not *define* the cross section of the infinite tree, your > > argument is empty. > > There is no need to define it. It is sufficient to know that EVERY > cross section is countable. No. That is not sufficient. When I ask you to define C(oo) you state that it is a cardinal number (namely aleph-0). By what way that will count the number of paths in the infinite tree escapes me. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: mueckenh on 9 Mar 2007 13:20 On 9 Mrz., 14:05, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <1173430532.626624.124...(a)c51g2000cwc.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > > > So, according to *your* definitions it is >. Not according to common > > > definitions. Would it not be possible that your definitions are not > > > consistent? It is your last paragraph which is inconsistent with the > > > other definitions. See how you *did* define: > > > > lim {n-->oo} |{1,2,3,...,n}| = aleph_0 > > > which means (as |{1,2,3,...,n}| = n: > > > lim {n-->oo} n = aleph_0? > > Do you not have a comment on this? It is wrong. A limit is either approached to any positive eps or it isn't a limit. lim {n-->oo} |{1,2,3,...,n}| = aleph_0 is only accepted as an assumption to be contradicted because set theory states that the set of all natural numbers has the cardinal number aleph_0. lim {n-->oo} n = aleph_0 is neither stated by set theory -nor by anyone else. > > No, they are *not*. The cross sections are sets of nodes (by your own > definition), except for C(oo), which is a cardinal number (again by > your own definition). And I see *no* relation between aleph-0 and > the paths. The cross section C(n) = |L(n)| is the number of nodes of the level L(n). > > > You have to provide that definition. There is also no standard > definition of limits that allows you to take the limit of cross sections > (which would be a set of nodes), so you have to provide that definition. > Finally, you have to *prove* that what the limit (by your definition) > gives is also the actual value. I have to show, an I have shown, that every finite Level L(n) is crossed by as many paths as are nodes in this level. This number of nodes is the countable cross section C(n) = |L(n)|. It is sufficient to have proved this in infinity, that is for EVERY level L(n), to show that the number of paths is countable as long as only nodes with finite indexes n contribute to the paths. If you disagree: What part of a path should not be covered by a node of a level L(n) with a finite n? Regards, WM
From: mueckenh on 9 Mar 2007 13:22 On 9 Mrz., 16:07, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <1173438655.923501.184...(a)8g2000cwh.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > > On 7 Mrz., 16:01, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > In article <1173215257.272668.121...(a)64g2000cwx.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > And this tells us precisely nothing about C(oo), which you *do* use. Forget it. Use only level L(n) and the countable number of nodes C(n) for every n in N to determine the number of paths crossin this level. If you find this insufficient, then tell me what after every n may be imagined. Regards, WM
From: Virgil on 9 Mar 2007 14:43
In article <1173437823.302617.229690(a)p10g2000cwp.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 7 Mrz., 22:57, Virgil <vir...(a)comcast.net> wrote: > > > > > C(oo) c |T(oo)|. (Both are cardinal numbers, both are aleph_0.) > > > > Then either represent the cardinality of the set of all paths in a > > complete infinite binary tree. > > > > C(n) is the cardinality of level L(n). There is no largest n. > Therefore this holds throughout the whole infinte tree. At no finite > position the infinite paths can be more than countably many. At each finite position corresponding to a finite tree, the set of infinite paths is empty. But that does not limit the set of infinite paths in an infinite tree to be empty, not to be countable. > > Either there is an infinite index. Or there are less than uncountably > many paths. Then WM must be claiming omega as an index. But it only indexes the length of the paths, not the "number" of them. > > > > > The limit lim {n-->oo} n is a number which n comes as close as you > > > like. > > > > Is this supposed to make sense? > > Otherwise limits do not make sense. WM's limits don't. > > > > > This is not omega or any greater number. > > > > Why not? > > > Because omega - n = omega > 1 Non sequitur. > > > > > > > > But you do *not* define the cross-section of the infinite tree, neither > > > > do you define the limits you are using. > > > > > I defined the cross section for every existing tree. > > > > No! Only for finite trees. > > For EVERY LEVEL which a paths can populate (if it can populate only > every finite level). If "it" can only "populate" finite levels, then WM is denying the existence of an infinite tree. > > > > > > > May be, but not in the tree --- as long as it stretches there is no > > > uncountable set. > > > > When one finally gets a complete infinite binary tree all "stretching" > > is over and done with, and one has all those uncountably many paths > > with nary a further stretch required. > > And after death you will enter the paradise and all pain will end and > you will no longer need any logics. But as long as you are here in the > vale of tears, there is logic dictating the facts. But WM has neither facts nor logic to back his arguments. In ZF and NBG, if there are any finite binary trees at all then there is also a complete infinite binary tree modeled by the set of all finite binary sequences as nodes and the set of infinite binary sequences as paths, and it has uncountably many paths. Since WM does not have any stated system in which he can prove his claims, they fail for lack of proof. |