Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: William Hughes on 15 Nov 2006 08:45 Franziska Neugebauer wrote: > William Hughes wrote: > > >> You want to say that A has a diagonal, but addition of one element to > >> each line, each column and the diagonal eliminates this property? > > > > Yes, because adding one element to each column changes the number > > of lines, but adding one element to each line does not > > change the number of columns. > > What did I tell you!? Yes, but the discussion has reached the point where the distinction between adding an element to an infinite sequence and adding an element to a finite sequence can be made clear. My contention is that an earlier attempt to clarify this would not have been productive. (not to say that this attempt will be productive, just that it is my opinion that it has a better chance of being productive). You may disagree, but it's my time that I'm wasting. > > [...] > > The original matrix has the same number of lines and columns, > > however,the lines and columns are different. Doing the same thing to > > the lines and columns does not have the same result. > > You are *not* doing the same thing! Yes and no. In both cases we are "adding a single element", in both cases we are increasing the ordinal represented by a sequence by one. So in one sense we are "doing the same thing". However, adding one element to a sequence without an end means we have to add the element "after" the sequence, while adding one element to a sequence with an end mean we add the element "at" the end. So in another sense we are not doing the same thing. My preferred informal description of this is "we are doing the same thing to two things that are different and we get different results". - William Hughes
From: Franziska Neugebauer on 15 Nov 2006 09:23 William Hughes wrote: > Franziska Neugebauer wrote: >> William Hughes wrote: [...] >> [...] >> > The original matrix has the same number of lines and columns, >> > however,the lines and columns are different. Doing the same thing >> > to the lines and columns does not have the same result. >> >> You are *not* doing the same thing! > > Yes and no. In both cases we are "adding a single element", in > both cases we are increasing the ordinal represented by a sequence > by one. In the "row case" you increase the finite number of occupied figures in each line which does neither change the domain of the sequence nor that of the matrix). In the "column case" you change the matrix structure by extending its domain. > So in one sense we are "doing the same thing". However, > adding one element to a sequence without an end means we have > to add the element "after" the sequence, while adding one > element to a sequence with an end mean we add the element > "at" the end. "Doing the same" applies solely to the wording not to the operations acting upon the underlaying mathematical objects. F. N. -- xyz
From: mueckenh on 15 Nov 2006 09:57 Dik T. Winter schrieb: > In article <1163505343.116057.183460(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > In article <1163428158.317887.311810(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > > > > Cantor considered well-ordering as a first principle, > > > > > > Zermelo introduced it at a first principle = axiom. Cantor was > > > > > > wrong, Zermelo was right? > > > > > > > > > > Cantor did state it without suggesting either that it was a first > > > > > principle or something else. He just assumed it. And he was wrong > > > > > with that assumption. > > > > > > > > You are wrong. "Der Begriff der wohlgeordneten Menge weist sich als > > > > fundamental für die ganze Mannigfaltigkeitslehre aus. Daß es immer > > > > möglich ist, jede wohldefinierte Menge in die Form einer > > > > wohlgeordneten Menge zu bringen, auf dieses, wie mir scheint, > > > > grundlegende und folgenreiche, durch seine Allgemeingültigkeit > > > > besonders merkwürdige Denkgesetz werde ich in einer späteren > > > > Abhandlung zurückkommen." (Cantor, Collected works, p.169) > > > > > > Ah, I missed that one. So he uses it as an axiom. > > > > Not in your sense. He wrote, for instance to Killing, on April 5, > > 1895: > > Was Herr Veronese darüber in seiner Schrift giebt, halte ich für > > Phantastereien und was er gegen mich darin vorbringt, ist unbegründet. > > > > Ueber seine unendlich großen Zahlen sagt er, daß sie auf anderen > > Hypothesen aufgebaut seien, als die meinigen. Die meinigen beruhen aber > > auf gar keinen Hypothesen sondern sind unmittelbar aus dem natürlichen > > Mengenbegriff abgezogen; sie sind ebenso nothwendig und frei von > > Willkür, wie die endlichen ganzen Zahlen. > > > > You see: Gar keine Hypothesen. Cantor's axioms are not chosen but they > > are necessary. > > Yes, he did regard it as such, that does not mean that he is right. And it does not mean that he is wrong. You cannot even decide this question without arbitrarily assuming axioms. Your choice is willful. Therefore your results do not prove anything and are without value. Cantor created absolute truth, at least where he avoided technical errors like his diagonal argument. > In > principle no axiom is necessary. But you need a few to have some start > to work with. That's the question. By means of axioms you can produce conditional truth at most. I am interested in absolute truth. Axioms will not help us to find it. I don't think we need any axioms. Regards, WM
From: mueckenh on 15 Nov 2006 10:09 Dik T. Winter schrieb: > In article <1163506214.782141.176790(a)m7g2000cwm.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > ... > > > > Thank you for your attention. Correction: > > > > An initial segment of natural numbers includes its cardinal number. > > > > > > Further correction. A finite initial segment of natural numbers includes > > > its cardinal number. > > > > Rejected. > > Why? Yes, I know why. You think that the complete set of natural numbers > does contain an un-natural number. It must, but cannot. As an un-natural number cannot be a natural number, this set N cannot exist. > I have asked you for a proof based on > the basic principles. But you simply refuse to do so. You think it is > the case, so it must be the case. So, please, a proof that N contains > something that is not a natural number based on the standard axioms of > set theory. Once you have done so you will have shown an inconsistency > because it is easy to proof that there is no such number. Here is one of several proofs: Take the set of natural numbers in form of a list or matrix: 1 11 111 .... This matrix has length omega and width omega. And its diagonal has length omega. No line has length omega. Therefore the width is larger than any line. And the diagonal is longer than any line. This is impossible. In order to see that, add one element to every column and to every line. Now the order type of any column is omega + 1, the length of the matrix has order type omega + 1, the order type of any line is n+1 < omega, and the width of the matrix has order type omega. The diagonal is a bijection between columns and lines. It des no exist. This shows that the diagonal in the original matrix did not exist either, unless there was a line of order type omega, i.e., an un-natural number. Regards, WM
From: mueckenh on 15 Nov 2006 10:36
Dik T. Winter schrieb: > > I don't. The parallel axiom is necessary, correct, true in any > > Euclidean plane. > > That is a tautology. I know. > But you do not allow elliptic or hyperbolic > geometry? If not, why not? I do not forbid it. It is clear that already the simple geometry on a sphere does not yield two parallels. Euclid simply did not consider such kind of plane. In another system we have other axioms (or better fundamental truths). > > I do not ask what you think. Reread my question. What is true about a > set of numbers in nature or reality? There are so many truths. Take order, 1 < 11, commutativity of addition and multiplication, n + m = m + n. These things do not become invalid or to be proved only because matrix multiplication or quaternions were invented. > > I think that the rise of non-euclidean geometries was a big > > disadvantage of mathematics, because from then on every guy felt > > entitled to create his own system of axioms. > > Before that time already, I think. And for those alternatives, blame > Saccheri, Lobachevsky and Riemann (resp. 1733, 1829 and 1854). But > consider the invention of the quaternions by Hamilton in 1843, where > he considered a kind of numbers that failed a basic field axiom. > > And still more basic: complex numbers. Failing the common ordering > axioms of numbers. Perhaps first used by Heron of Alexandria (1st > century CE). But first actually used in 1545 in the book by Cardano > (but that was credited to Tartaglia). > > Of course, at that time people did not thoroughly think in strict > axiomatic systems, but in what they thought were fundamental truths. > But in essence there is no difference. I think there is a great difference. It is not necessary to call negative solutions "false" solutions as even Descartes did, (because it was customary at his time. Although this custom was justified as long as only positive numbers were called numbers.) But it is necessary to distinguish between negative and positive numbers or real and complex numbers or Euclidean and non Euclidean spaces. Then it will not be astonishing that some new axioms arise or some existing axioms have to be modified. Going from addition to subtraction, we find that commutativity is no longer valid. And the natural numbers are not closed under subtraction. But these adaptions of axioms (or I should better say: of fundamental truths) has to be justified and not only to be decided by an arbitrary choice. Regards, WM |