Cantor Confusion
in [Math]
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From: Virgil on 9 Mar 2007 14:58 In article <1173438655.923501.184140(a)8g2000cwh.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. Is that a "royal" "we"? It is certainly not a plural one. > > > > 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. WM mutters his own creed without proofs, whereas the "creed" that the set of binary infinite sequences is uncountable has a valid proof. > > > > 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. There is if one is going to make claims about it as WM keeps doing. > It is sufficient to know that EVERY > cross section is countable. It is sufficient to know that 2 + 2 = 5 in order to have all of WM's claims become provable. But unfortunately 2+2 is not 5, and few, if any, of WM's claims are provable.
From: Virgil on 9 Mar 2007 15:00 In article <5ug2v215gvfu0rblhvg6jdff7092nr81f2(a)4ax.com>, G. Frege <nomail(a)invalid> wrote: > On 9 Mar 2007 03:10:56 -0800, mueckenh(a)rz.fh-augsburg.de wrote: > > >> > >> As you do not *define* the cross section of the infinite tree, > >> your argument is empty. > >> > > There is no need to define it. [...] > > > > A very good example of M�ckenmathics! :-) > I thought it was M�ckenmatics.
From: Virgil on 9 Mar 2007 15:10 In article <1173464401.633116.124900(a)q40g2000cwq.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. What does " approached to any positive eps" mean when translated into mathematically coherent language? It certainly has no mathematically sensible meaning as is. > > 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. As it is only your claim which is being contradicted, and not a claim made by mathematics, it is only you who face such contradiction. > > 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). Which is only the case when n is a natural number. > > > > > > 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. That has long since been agreed upon by everyone. What is not agreed upon, and is not true, is that any part of that carries over to non-naturals. 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), But it is also necessary to show that it holds for non-natural cardinals or ordinals, and this has not been done. > > If you disagree: What part of a path should not be covered by a node > of a level L(n) with a finite n? The square root of Newton's apple. Not that my answer makes as much sense as your question. > > Regards, WM
From: Virgil on 9 Mar 2007 15:15 In article <1173464550.644857.271850(a)30g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. For every function, f, from N to {m,w}, there is a corresponding infinite path whose nth branch is to the left child if f(n) = m and to the right child if f(n) = w, and vice versa. Cantor proved that the set of all such function from N to {m,w} is uncountable. If WM wants anyone to believe that the set of all corresponding infinite paths is countable, he will have to find a way to actually count them, i.e., find a concrete surjection from N to the set of all such paths.
From: mueckenh on 10 Mar 2007 06:09
On 9 Mrz., 20:43, Virgil <vir...(a)comcast.net> wrote: > In article <1173437823.302617.229...(a)p10g2000cwp.googlegroups.com>, > > > > > > 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. Does an infinite tree possess any level which is *not* enumerated by a finite natural number n, i.e., which has an infinite distance to te tree? Regards, WM |