Cantor Confusion
in [Math]
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From: mueckenh on 3 Mar 2007 03:32 On 1 Mrz., 22:59, Franziska Neugebauer <Franziska- Neugeba...(a)neugeb.dnsalias.net> wrote: > mueck...(a)rz.fh-augsburg.de wrote: > > On 1 Mrz., 11:45, Franziska Neugebauer <Franziska- > > Neugeba...(a)neugeb.dnsalias.net> wrote: > >> mueck...(a)rz.fh-augsburg.de wrote: > >> > 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. > > >> There is no objective in mathematics to "unv[e]il" physical > >> (material) restrictions of the world. > > > After having recognized them > > Mission not yet accomplished. Please keep posting on until _we_ have > "recognized them". Good mathematicians did recognize this already 100 years ago. Here is a quote from Koenig, 1905 (a memorable year in several respects). Es wird vor allem die ,,Tatsache" angenommen, daß es in unserem Bewußtsein sich abspielende Prozesse gibt, die den formalen Gesetzen der Logik genügen und als ,,wissenschaftliches Denken" bezeichnet werden, und daß es unter diesen auch solche gibt, die mit anderen ebensolchen Prozessen, der Erzeugung jener früher beschriebenen Reihenfolgen, in gegenseitig eindeutiger Beziehung stehen. Die Frage, ,,wie" diese Beziehung zustande kommt, oder gar ,,wie weit" diese Beziehungen erstreckt werden können, wird dabei gar nicht berührt. (Metalogisches Axiom.) Regards, WM
From: mueckenh on 3 Mar 2007 05:22 On 3 Mrz., 02:02, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <1172774506.011332.315...(a)p10g2000cwp.googlegroups.com> mueck....(a)rz.fh-augsburg.de writes: > > > On 1 Mrz., 01:48, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > ... > > > > > > You know that every natural number like 2n is finite and therefore > > > > > > less than aleph_0 while |{2,4,5,...,2n}| in the limit is aleph_0. > > > > > > > > > > And 2n in the limit is aleph_0, when you define such limits. > > > > > > > > That is what I say, but according to set theory it is wrong. > > > > > > Where is that stated? Set theory does not define limits, so it is not > > > stated in set theory. > > > > Every nondecreasing sequence of real numbers bounded from above has a > > limit. There is a limit superior and a limit inferior proven by set > > theory. > > Not defined by set theory. Karel Hrbacek and Thomas Jech: "Introduction to Set Theory" Marcel Dekker Inc., New York, 1984, 2nd edition. 3.8 Theorem Every bounded sequence of real numbers has a convergent subsequence. The number a = sup {inf {ak | k ï³ n} | n ï N} used in the proof of Theorem 3.8 is called the limit inferior of ï¡anï±. Similarly b = inf {sup {ak | k ï³ n} | n ï N} is called the limit superior of ï¡anï±. Prove that (a) b exists for every bounded sequence and a ï£ b. Of course definitions and proofs are done by set theory > > > In order to get to higher ordinals, Hrbacek aqnd Jech say: It is easy > > to continue the process after this "limit" step is made. > > The meaning of 'limit' here is soemthing different, that is why it is in > quotes. > Yes, it is an extension, like the irrational "numbers" are an extension of the natural numbers. The latter quotation marks are not necessary, according to you. Wh yare the former? > > In fact there are limits in set theory. "Es ist sogar erlaubt, sich > > die neugeschaffene Zahl omega als Grenze zu denken, welcher die Zahlen > > nu zustreben, wenn darunter nichts anderes verstanden wird, als daà > > omega die erste ganze Zahl sein soll, welche auf alle Zahlen nu folgt, > > d. h. gröÃer zu nennen ist als jede der Zahlen nu." > > No there are no limits defined in set theory. See above. > What this states it that > you can look at it as such, but that it is not such, because limits are > not defined. They are not well defined. I agree. I snip the rest because our discussion should be focused on the more relevant topics, the tree in particular. Regards, WM
From: mueckenh on 3 Mar 2007 05:36 On 3 Mrz., 02:09, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <1172834875.244086.99...(a)31g2000cwt.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > > Physical restrictions are not something mathematicians are concerned > > > with. > > > > Environmental questions were not something politicians were concerned > > with. This is now changing. > > This has already been changing a long time. But I fail to see the > relevance. > The consciousness of the politicians has begun to change only very recently. The same will happen with the mathematicians - before all available bits will have been used up, I hope. > > pray start a form of mathematics that *does* use it. Why should I start it? All mathematicians are forced to use it because there is no alternative. Alas most of them apply it without knowing that they apply it. > But do not use it to show contradictions in current mathematics if they are > only contradictions with MatheRealism. > > > > > > 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). > > But here you use your own personal language to state what mathematicians > state is wrong. So you are teaching your pupils something wrong. Look, I teach physics too, mostly Newtonian physics, only few things from Einstein or Heisenbeg. Am I lying? Does an engineer who is expected to construct gearboxes need relativity? > > You are trying to avoid stubborness? Not only, I even succeed. Regards, WM
From: Virgil on 3 Mar 2007 05:49 In article <1172918163.656673.294060(a)30g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > You are trying to avoid stubborness? > > Not only, I even succeed. Like many of your other claims, not proven, and claimed despite strong contrary evidence.
From: mueckenh on 3 Mar 2007 05:52
On 3 Mrz., 02:18, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <1172835359.498019.275...(a)64g2000cwx.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > > On 2 Mrz., 04:44, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > > 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 number of unit > > fractions in the last parenthesis used by Oresme. > > I missed something. What is the definition of the cross section of the > "unit tree"? What is the definition of the "unit tree"? What is the definition of the last parentheses of the proof of the divergence of the harmonic series? It is not defined, but all parentheses are required to exist. The same is valid for the cross sections of the trees. > > > > > > 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))) > > So you refuse to define it? As long as you do not define it, it can not be > ascertained whether the statement above is true or false. The union of all trees is defined in your opinion? The cross section C(oo) can be defined as the limit of all finite cross sections. It can be defined as the cross section of the union tree U(T(n)). It can also be defined as the cross section of the of the complete tree T(oo). If we use the definition of C(oo) as the cross section of the union tree U(T(n)), then we have the following facts: The tree T(n) can mapped on the natural number of its nodes. There are infinitely many trees in the union U(Tn)) as there are infinitely many natural umbers in the union N. The union has a finite size concerning the sizes of all natural numbers. None of them is infinite. Therefore the nodes of the union tree have a finite cardnality. (Recall every union of finite trees is the largest tree in the union.) If we use the definition of C(oo) as the cross section of T(oo) then we have the estimation: The cross section cannot surpass the number of all nodes. This number is coutable, hence C(oo) =< Card(T(oo)) = aleph_0. > Regards, WM |