Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: Dik T. Winter on 9 Nov 2006 20:13 In article <1163068762.173696.311700(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > How wrong you are. In the case of the edges you show that for finite trees > > of order n: > > #edges(n) = f(n) > > with some function n. From that you conclude: > > #edges(oo) = f(oo) > > which requires transfinite induction. To prove that Q is countable you do > > not need infinity at all. See the construction of a bijection I gave > > from N to Q+, using simplified continued fractions. This shows that > > every finite n maps to a finite q>0 and the reverse. So we have a > > function g: N -> Q+ that has an inverse. And that is all that is needed > > to show countability. Where does the tranfinite induction come in? > > There is no difference between enumerating all the rational numbers in > the well-known scheme, starting at the corner of an infinite square > matrix There is. You can assign numbers to edges that terminate at nodes. And *that* numbering is different. It numbers only those edges that are finitely far away from the root. But that way you do not number all edges in the infinite tree, because that contains edges that are *not* finitely far away from the root. > > Indeed. To prove countability you do not need transfinite induction. > > But my remark "you are wrong" was to your statement: > > "The mapping N -> {edges} has been established". > > See J. H. Conway, On Numbers and Games, for a clear exposition about the > > difference between the union of all finite trees and the infinite tree. > > Whatever Conway may say, there is no difference between enumerating all > rational numbers in the well-known scheme, starting a the corner of an > infinite square matrix, and the edges of the tree. In both cases you > have a system with a limit which is countably infinite. There is a difference, see above. You can only number edges that are finitely far away from the root, but in that way you will only number edges that terminate at nodes finitely far away from the root. As in the edges there are nodes infinitely far away from the root you will never number the edges that terminate there. The situation with the rationals is quite different, because in the matrix *each* rational is finitely far away from the root. > Perhaps you see a difference between the union of all finite numbers n > and the set N? Apparently you are not interested in what Conway did write, otherwise you would understand how ridiculous this comment is. -- 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 9 Nov 2006 21:03 In article <1163069363.908618.240110(a)h54g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > > Perhaps. But we do not talk here about length or weight. We are > > > > talking mathematics. > > > > > > How else lengths and weight could be compared if not by mathematics? > > > > Weights by scales. Lengths by putting them alongside. No mathematics > > involved. > > Comparison of numbers by size is one of the first tasks of mathematics. You think so. As I said above, comparing weights and lengths does not involve mathematics. > > But at least weight is not a mathematical entity. Length is, > > but not in the way you see it. > > The numbers counting the units of weight or length are mathematical > entities. They are. But in mathematics comparing numbers comes in quite late in the process of definition. It starts when the ordering axioms are given. In a set a ardering relation may be can be defined. Let us use >= as the basis. This defines an ordering relation if it is (1) reflexive, i.e. if a >= a (2) anti-symmetrics, i.e. if a >= b and b >= a then a = b (3) transitive, i.e. if a >= b and b >= c then a >= c and when the set is considered a ring: (4) invariant to translation, i.e. if a >= b then a + c >= b + c for every c (5) invariant to scaling, i.e. if c >= 0 and a >= b then a.c >= b.c Note that for the complex numbers, the relation: a >= b if a_r > b_r or (a_r = b_r and a_i >= b_i) is an ordering relation when they are considered as a set, but not when they are considered as a ring. But set theory is from the basis not interested in ordering. At some point (once you have define cardinality) you can define an ordering relation between cardinalities: |A| >= |B| if there is a surjection from A to B. This is easily shown to be an ordering relation. How much more do I need to tell you about the basics of mathematics? This is all part of the first year (or perhaps the first two years) for a student in mathematics at university. > > > Why didn't you? > > > > I did. Why do you think I did not? > > Because you do not know that in ZFC everything is a set, In some *models* of ZFC everything is a set. In some models that is not the case. > becauses you > do not know that limits are inside set theory. Where are they? That there are "limit ordinals" does not mean 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 da? omega die erste ganze > Zahl sein soll, welche auf alle Zahlen nu folgt, d. h. gr??er zu > nennen ist als jede der Zahlen nu" [Cantor, Collected works, p. 195]. Since Cantor quite a bit has changed. Limits are *not* part of set theory, they belong to topology and other things build on set theory. I think I have followed more courses on set theory than you did. I, at least, did pass my examination on set theory. Did you when you were at university? The second quote tells me *nothing* about either the definition of limits, nor about everything being a set. But if you want to start with set theory with Cantors methods, go ahead. You have still a few years of development in set theory in front of you. Try to do physics with late 20th century physics. You may get along a long way, but you will find limits as you proceed. -- 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 9 Nov 2006 21:41 In article <1163098294.339005.143210(a)e3g2000cwe.googlegroups.com> "MoeBlee" <jazzmobe(a)hotmail.com> writes: > mueckenh(a)rz.fh-augsburg.de wrote: .... > > Es soll nun gezeigt werden, wie man zu den Definitionen der neuen > > Zahlen gef?hrt wird > > It is now to be shown how one is lead to the definition of the new > > numbers. And this definition, given by Cantor and translated by myself > > is given above. There is no point to draw but only to understand > > Cantor's *definition* (or not). > > Okay, so for you there is no point to draw about your quote other than > understanding Cantor's own writings. Indeed. Wolfgang Mueckenheim appears to think that Cantor's writings are still all true in modern mathematics. He fails to see that although his thinking form the basis, not all his observations are still true, or, perhaps, were never true. He just started a new way of thinking. The first falseness is his assumption that every set can be well-ordered. We all know now that that can not be proven from first principles. His second falsehood is when he states that a set of first cardinality (meaning sets of cardinality aleph-0) can only be counted with use of numbers of the second class (meaning omega and larger). And I think that especially this quote has lead Wolfgang Mueckenheim astray. Also see my discussion about this quote with Dave Seaman. The falsity is apparent if you realise that quote means that every set with cardinality aleph-0 has an omega-th element. I think this has to do with the problems of the conceptions of potential infinity vs. actual infinity that played an important role in mathematics before the early 1900's. Moreso because there were also religious issues. So Cantor crossed the border (after consultation with his religious leader and in strong opposition from him). The actual infinite was impossible (and unreligious) before that time, but he introduced actual infinity. And so, apparently, he thought that if a set was actually infinite, it should have an actually infinite element, or somesuch. He failed to see that, while the contents of a set could be "potentially" infinite (i.e. unbounded), the size of such a set could be "actually" infinite. What surprises me is that opposition against it in this way comes from especially the German posters in this group. They all want to stress the difference between actual infinity and potential infinity, that are indeed not mathematical terms, and most other posted do not know the difference. Is it still the old religious issue? I think so, based on the postings of at least one of the German posters. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: David Marcus on 9 Nov 2006 23:23 Dik T. Winter wrote: > In article <1163068502.005252.83170(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > In article <1162820871.886446.129490(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > > No, not mixing at all. Supremum only. > > > > > > > > Then 0.111... is not different from the finite sequences of 1's? > > > > > > How do you conclude that? Given finite initial segments of the triangle > > > we get a length, a width and a length of the diagonal that are all equal. > > > When we look at the complete triangle (which exists by the axiom of > > > infinity) all three become the supremum of the finite quantities, and > > > so still are equal. No maximum involved. > > > > > The diagonal and the left side of triangle are complete with infinitely > > many ones, but no line has infinitely many 1's. > > And, what exactly is the problem? WM's logic (if we can call it that) seems to be that for a finite triangle, the length of the diagonal is the same as the length of the last line. Therefore, this should be true for infinite triangles, too. Why he would think that something that is true for finite things should automatically be true for infinite things is a mystery to me. -- David Marcus
From: mueckenh on 10 Nov 2006 05:11
William Hughes schrieb: > > There are two sets the bijection of which has not yet been considered > > but sould be: > > A = The set of lengths of initial segments of the (first) column. > > B = The set of lines. > > Here we can set up a bijection, expressing the lines by the natural > > indexes of the elements: > > > > 1 1 > > 2 1,2 > > 3 1,2,3 > > ... ... > > n 1,2,3,...,n > > ... ... > > omega 1,2,3,... > > > > which shows that omega or aleph_0 does not exist. > > No. This shows that there is no line omega. The number > of lines is omega. This does not mean that there is > a line omega. Your assertion: "The number of lines is omega. This does not mean that there is a line omega" contains a self-contradiction. Consider the bijection between initial segments of columns and lines like this typical example 1 2 3 .... n <--> 1,2,3,...n The lines are mapped on finite segments of the column only. There is no line with a finite number of elements mapped on an initial segment with omega elements. Therefore the number of lines is finite only, but not infinite. Therefore there are less than omega lines. As a consequence an actually existing number omega > n cannot exist. Regards, WM |