Cantor Confusion
in [Math]
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From: Dik T. Winter on 12 Mar 2007 11:41 In article <1173464550.644857.271850(a)30g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > 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. The paths in finite trees terminate at soe level n. The paths in the infinite tree do not terminate. Do you not see the difference? -- 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 12 Mar 2007 12:01 On 12 Mrz., 16:41, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <1173464550.644857.271...(a)30g2000cwc.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > > 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. > > The paths in finite trees terminate at soe level n. The paths in the > infinite tree do not terminate. Do you not see the difference? > -- The levels which have a countable number of nodes do not terminate either. Therefore there is no difference: As long as paths can exist the cardinality of them is restricted to a countable number. An uncountable set of paths cannot exist other than outside of the tree, i.e., outside of mathematics. Regards, WM
From: mueckenh on 12 Mar 2007 12:08 On 11 Mrz., 17:21, "Gc" <Gcut...(a)hotmail.com> wrote: > 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. Sorry if I have I have insulted you, but I have mixed you up with someone else. My error. To answer your question: If there are single paths in the tree, then their set is subject to the restriction imposed by the cross sections (= numbers of nodes) of the levels. These single paths cannot exist outside of any level. Every level has a countable number of nodes. If there are no single paths in the tree, ... then: where are they? Regards, WM
From: mueckenh on 12 Mar 2007 14:34 On 12 Mrz., 16:40, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <1173464401.633116.124...(a)q40g2000cwq.googlegroups.com> mueck...(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. It is not my definitin, but it is the definition of set theory. Of course it is wrong. > > > 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. I read that book. Therefore I know their definitions and I knew already that set theory works with limits when you disputed that. Nevertheless their definition does not apply to the infinite set of finite numbers. (Further it is wrong because what they "call the limit" is not a limit. But this is irrelevant here.) The actually infinite set of finite numbers (without their supremum) is by set theory defined as: lim {n-->oo} |{1,2,3,...,n}| = aleph_0 (that expresses the actually infinite set) lim {n-->oo} n < aleph_0 (that is because every number is finite). > > > > 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. aleph_0 is an upper bound for any set of separated paths at a finite level. Shopuld there be more paths, then they had to cross at least one level L(alpha) with an infinite number alpha. > > > > 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. No. The infinite paths do not terminate at any level. They cross the level L(n) and then there are 2^n paths (-bundles). In any finite evel there are countably many paths (-bundles). > 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? No. Up to the level L(1) there are only two separated path (-bundles). Up to level L(n) ther are 2^n. And only if there are levels with non- natural number there can be more than countably many paths (-bundles). Regards, WM
From: Virgil on 12 Mar 2007 17:10
In article <1173715319.243468.87220(a)c51g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 12 Mrz., 16:41, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article <1173464550.644857.271...(a)30g2000cwc.googlegroups.com> > > mueck...(a)rz.fh-augsburg.de writes: > > > > > 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. > > > > The paths in finite trees terminate at soe level n. The paths in the > > infinite tree do not terminate. Do you not see the difference? > > -- > > The levels which have a countable number of nodes do not terminate > either. Therefore there is no difference: As long as paths can exist > the cardinality of them is restricted to a countable number. Then the alleged "union" of those infinitely many finite trees does not produce the complete infinite binary tree, which has been repeatedly proved to have uncountably many paths. > An uncountable set of paths cannot exist other than outside of the > tree, i.e., outside of mathematics. Inside of mathematics but outside of WMatics. The complete infinite binary tree exists within both ZF and NBG, both of which are inside of mathematics but outside of WMatics. Perhaps one day WM will be able to distinguish between his WMatics and actual mathematics. But apparently not today. |