Cantor Confusion
in [Math]
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From: Dik T. Winter on 1 Mar 2007 22:34 In article <1172740041.611220.208880(a)8g2000cwh.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > On 1 Mrz., 01:22, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article <1172650983.733643.316...(a)m58g2000cwm.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > > > What is misleading about it if every mathmatician calls them "numbers"? > > It veils the physical restrictions. Physical restrictions are not something mathematicians are concerned with. > > sqrt(2) is in trichotomy with the rational numbers. > > In fact not even all rational numbers are numbers. You know, there are > less than 10^100 bits ... In that case not all natural numbers are numbers. And being a number depends on the whim of the observer. But that is only your finitistic approach. > > So in your mathematical courses you call what we call the > > "complex numbers", "complex ideas" or something like that? > > I use the current words. I do not talk about these things there. So you are lying (in your opinion) to your pupils? > > At least those are not in > > trichotomy with the "really real numbers". > > Yes. I would prefer another word, but that will be impossible to > introduce. You may have noticed that I talk of "irrationale Zahl" even > in my book. I had considered to use only "Irrationalit�t". But it > sounds too heavy-handed. Not only that. -- 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 1 Mar 2007 22:44 In article <1172741488.047161.76960(a)h3g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > On 1 Mrz., 01:28, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article <1172651060.321953.253...(a)v33g2000cwv.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > > As all the finite subsets like {p(0), p(1), p(2), ...} which belong to > > > an infinite path like p(oo) form a countable set, the set of all > > > unions is countable. This set of all unions is P(oo) = {p(oo), q(oo), > > > ...}. > > > > Proof, please. Given the set of paths. Each subset of that set defines > > a union of paths. > > Not every subset of the set of paths defines a union which is an > infinite path. There are only very few such subsets of he set of > paths. Yes, where did I state something different? Do not argue to me about things I do not state, please. > Here we have the same situation as with the number 2^omega which has > cardinality aleph_0. There is a one-to-one map to the cross sections > of the trees and hence to the cross section of the unit tree. No, you have not. Review the definition of 2^omega. It is only valid when that cross section of the unit tree can be well-ordered. Pray show a well-ordering of the cross section of the unit tree. (Ordinal numbers require well-ordering.) > > > It is easy to understand. We are investigating this assertion. > > > We find that every finite tree contains only finite paths. > > > The union of some finite paths is an infinite path. > > > Thee are only countable many unions of finite paths which > > > yield infinite paths. > > > > A proof, please, that there are only countably many unions of finite paths > > which yield infinite paths. > > The set of unions cannot be larger than the cross section of the tree > U(T(n)) which is Card(2^omega). Proof please that that cross section is well ordered. Anyhow, a cross section that you refuse to define? > > But what are you now stating, that those > > infinite paths are *not* in the infinite tree? Where are they? > > I said: The union of some finite paths is an infinite path. There are > only countable many unions of finite paths which > yield infinite paths You stated something different. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Virgil on 2 Mar 2007 00:48 In article <1172793888.468366.159710(a)30g2000cwc.googlegroups.com>, "David R Tribble" <david(a)tribble.com> wrote: > Han de Bruijn wrote: > >> Do you deny that Mathematics is supposed to be a no-nonsense discipline? > > > > Randy Poe wrote: > > Lewis Carroll, a mathematician, was fond of playing > > these semantic games but that doesn't make it mathematics. > > > > 1. Tyrannical governments are arbitrary. > > 2. Let A be an any arbitrary government. > > 3. Then A is tyrannical. > > Then there is: > 1. Caterpillars eat cabbage. > 2. Han eats cabbage. > 3. Therefore Han is a caterpillar. With Mueckenheim as cabbage?
From: mueckenh on 2 Mar 2007 06:27 On 2 Mrz., 04:34, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <1172740041.611220.208...(a)8g2000cwh.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > > On 1 Mrz., 01:22, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > In article <1172650983.733643.316...(a)m58g2000cwm.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > > > > > What is misleading about it if every mathmatician calls them "numbers"? > > > > It veils the physical restrictions. > > Physical restrictions are not something mathematicians are concerned with. Environmental questions were no something politicians were concerned with. This is now changing. > > > > sqrt(2) is in trichotomy with the rational numbers. > > > > In fact not even all rational numbers are numbers. You know, there are > > less than 10^100 bits ... > > In that case not all natural numbers are numbers. And being a number depends > on the whim of the observer. But that is only your finitistic approach. It is he MatheRealism. Finitism is not well defined. There are finitists who claim a largest number, others do not. Further there is a negative after taste with finitism. Realism will have better chances to succeed. > > > > So in your mathematical courses you call what we call the > > > "complex numbers", "complex ideas" or something like that? > > > > I use the current words. I do not talk about these things there. > > So you are lying (in your opinion) to your pupils? No, I use another language. And I do not go into a depths the students would not understand (and not need). > > > > At least those are not in > > > trichotomy with the "really real numbers". > > > > Yes. I would prefer another word, but that will be impossible to > > introduce. You may have noticed that I talk of "irrationale Zahl" even > > in my book. I had considered to use only "Irrationalität". But it > > sounds too heavy-handed. > > Not only that. Oh, "Irrationalität" is used in German synonymously to "Irrationalzahl", but using it exclusively would show some inflexibility and stubbornness which I always try to avoid. Regards, WM
From: mueckenh on 2 Mar 2007 06:35
On 2 Mrz., 04:44, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <1172741488.047161.76...(a)h3g2000cwc.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > > On 1 Mrz., 01:28, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > In article <1172651060.321953.253...(a)v33g2000cwv.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > > > As all the finite subsets like {p(0), p(1), p(2), ...} which belong to > > > > an infinite path like p(oo) form a countable set, the set of all > > > > unions is countable. This set of all unions is P(oo) = {p(oo), q(oo), > > > > ...}. > > > > > > Proof, please. Given the set of paths. Each subset of that set defines > > > a union of paths. > > > > Not every subset of the set of paths defines a union which is an > > infinite path. There are only very few such subsets of he set of > > paths. > > Yes, where did I state something different? Do not argue to me about > things I do not state, please. > > > Here we have the same situation as with the number 2^omega which has > > cardinality aleph_0. There is a one-to-one map to the cross sections > > of the trees and hence to the cross section of the unit tree. > > No, you have not. Review the definition of 2^omega. It is only valid > when that cross section of the unit tree can be well-ordered. Pray show > a well-ordering of the cross section of the unit tree. (Ordinal numbers > require well-ordering.) The cross section of the tree is the same as the numkber of unit fractions in the last parenthesis used by Oresme. Well-ordering of all unit fractions is as easy as well-ordering of all nodes enumerated by natural numbers. But in order to save you from showing that 2^omega is countable, simply consider that the cross section of U(T(n)) cannot be larger than the set of all nodes of U(T(n)). And that set is countable. > > > > > It is easy to understand. We are investigating this assertion. > > > > We find that every finite tree contains only finite paths. > > > > The union of some finite paths is an infinite path. > > > > Thee are only countable many unions of finite paths which > > > > yield infinite paths. > > > > > > A proof, please, that there are only countably many unions of finite paths > > > which yield infinite paths. > > > > The set of unions cannot be larger than the cross section of the tree > > U(T(n)) which is Card(2^omega). > > Proof please that that cross section is well ordered. Anyhow, a cross > section that you refuse to define? C(oo) =< Card(U(T(n))) > > > > But what are you now stating, that those > > > infinite paths are *not* in the infinite tree? Where are they? > > > > I said: The union of some finite paths is an infinite path. There are > > only countable many unions of finite paths which > > yield infinite paths > > You stated something different. ? But I meant what I said above. Regards, WM |