Cantor Confusion
in [Math]
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From: mueckenh on 11 Mar 2007 03:10 On 11 Mrz., 01:52, Virgil <vir...(a)comcast.net> wrote: > In article <1173563630.875399.52...(a)p10g2000cwp.googlegroups.com>, > > mueck...(a)rz.fh-augsburg.de wrote: > > On 10 Mrz., 18:03, Virgil <vir...(a)comcast.net> wrote: > > > In article <1173524975.470534.275...(a)q40g2000cwq.googlegroups.com>, > > > > mueck...(a)rz.fh-augsburg.de wrote: > > > > 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? > > > > Does the infinite set of naturals need to contain an infinite natural? > > > No more does an infinite set of levels need to contain an infinite level. > > > Therefore it suffices to show that the number of paths of the tree is > > countable for every level L(n) which is enumerated by a finite number. > > This proof shows that all path in the tree are countable- > > it only shows that the set of those paths which have a "level" , i.e., > end, are countable. Nonsense. The cross section does not measure "infinite paths which have an end". > It says nothing at all about the number of endless > paths. Every level L(n) measures, by |L(n)|, how many paths are present at level n. Obviously at each level only a finite set of paths is present. As the sequence of levels does never end, there is an unending growth of this number but it remains provably countable. That is the meaning of infinite paths in an infinite tree. Regards, WM
From: Virgil on 11 Mar 2007 03:27 In article <1173597014.002770.146430(a)q40g2000cwq.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 11 Mrz., 01:52, Virgil <vir...(a)comcast.net> wrote: > > In article <1173563630.875399.52...(a)p10g2000cwp.googlegroups.com>, > > > > mueck...(a)rz.fh-augsburg.de wrote: > > > On 10 Mrz., 18:03, Virgil <vir...(a)comcast.net> wrote: > > > > In article <1173524975.470534.275...(a)q40g2000cwq.googlegroups.com>, > > > > > > mueck...(a)rz.fh-augsburg.de wrote: > > > > > 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? > > > > > > Does the infinite set of naturals need to contain an infinite natural? > > > > No more does an infinite set of levels need to contain an infinite > > > > level. > > > > > Therefore it suffices to show that the number of paths of the tree is > > > countable for every level L(n) which is enumerated by a finite number. > > > This proof shows that all path in the tree are countable- > > > > it only shows that the set of those paths which have a "level" , i.e., > > end, are countable. > > Nonsense. The cross section does not measure "infinite paths which > have an end". It does not count the set of infinite paths at all, as it is only defined for finite trees. > > > It says nothing at all about the number of endless > > paths. > > Every level L(n) measures, by |L(n)|, how many paths are present at > level n. Obviously at each level only a finite set of paths is > present. As the sequence of levels does never end, there is an > unending growth of this number but it remains provably countable. Does that argument work with the set of terminating decimal or terminating binary fractions being extended to infinite? No, because in both cases the infinite cases are uncountable. WM still has not been able to fault card(S) < card(P(S)), which, in view of the validity of its proof is, not surprising. Nor has he been able to fault my bijection between the set of all infinite paths of an infinite binary tree and the uncountable set P(N).
From: mueckenh on 11 Mar 2007 04:33 On 11 Mrz., 00:29, "Gc" <Gcut...(a)hotmail.com> wrote: Please stick to one pseudonym. Otherwise it is tedious to killfile you. I don't want to read you. Regards, WM
From: Gc on 11 Mar 2007 12:21 On 11 maalis, 10:33, mueck...(a)rz.fh-augsburg.de wrote: > On 11 Mrz., 00:29, "Gc" <Gcut...(a)hotmail.com> wrote: > > Please stick to one pseudonym. Otherwise it is tedious to killfile > you. I don't want to read you. > > Regards, WM I don`t have other pseudonyms, but fine: I will leave you alone with your hallucinations.
From: Dik T. Winter on 12 Mar 2007 11:40
In article <1173464401.633116.124900(a)q40g2000cwq.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > 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. We are not doing analysis here. But you now state that your definition: lim {n->oo} |{1,2,3,...,n}| = aleph_0 is wrong? Strange as you use it. > 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. This makes no sense. You provided it as a *definition*. If not, how do you *define* that thing? > lim {n-->oo} n = aleph_0 is neither stated by set theory -nor by > anyone else. See page 193 of Hrbacek and Jech where you will find a definition of limit from which you can derive precisely that. > > 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). I still see no relation between aleph-0 and the paths. > > 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. But you have not proven that. You have only proven that there are as many paths that *terminate* at nodes at that level as there are nodes at that level. Each node is crossed by more than one path. Actually each node is crossed by uncountably infinite many paths. > 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? If you disagree, are there only two paths in the tree because the cross-section C(1) contains only two nodes? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |