Another AC anomaly?
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From: Virgil on 27 Nov 2009 18:33 In article <e998c756-bb05-4370-be93-5f811aee443a(a)a21g2000yqc.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 27 Nov., 18:55, A <anonymous.rubbert...(a)yahoo.com> wrote: > > On Nov 27, 12:36�pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > > > > > > > > > On 27 Nov., 16:21, A <anonymous.rubbert...(a)yahoo.com> wrote: > > > > > > On Nov 27, 1:43�am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > > > On 27 Nov., 03:50, A <anonymous.rubbert...(a)yahoo.com> wrote: > > > > > > > > On Nov 26, 3:51�pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > > > > > On 26 Nov., 19:22, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > > > > > > On Nov 26, 12:24�pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > > > > > > > Here is another interesting task: Use balls representing the > > > > > > > > > positive > > > > > > > > > rationals. The first time fill in one ball. Then fill in > > > > > > > > > always 100 > > > > > > > > > balls and remove 100 balls, leaving inside the ball > > > > > > > > > representing the > > > > > > > > > smallest of the 101 rationals. > > > > > > > > > > [at random with any measure that gives a positive probability > > > > > > > > to each rational] > > > > > > > > > Simply take the first, seconde, third ... Centuria according to > > > > > > > Cantor's well-ordering of the positive rationals. Then there is > > > > > > > no > > > > > > > need for considering any probabilities. > > > > > > > > > > > If you get practical experience, you > > > > > > > > > will accomplish every Centuria in half time. So after a short > > > > > > > > > while > > > > > > > > > you will have found the smallest positive rational. > > > > > > > > > > Only in Wolkenmuekenheim. �Outside of Wolkenmuekenheim > > > > > > > > you will have an empty set. > > > > > > > > > Besides your assertion, you have arguments too, don't you? > > > > > > > In particular you can explain, how the empty set will emerge > > > > > > > while > > > > > > > throughout the whole time the minimum contents of the vase is 1 > > > > > > > ball? > > > > > > > > > Regards, WM > > > > > > > > Let S denote a set with exactly 101 elements. Let Q+ denote the > > > > > > positive rational numbers. Let inj(S,Q+) denote the set of > > > > > > injective > > > > > > functions from S to Q+. Let {x_n} denote a sequence of elements of > > > > > > inj > > > > > > (S,Q+) with the following properties: > > > > > > > > 1. Let im x_n denote the image of x_n. Then the union of im x_n for > > > > > > all n is all of Q+. > > > > > > > > 2. For any n, the intersection of im x_n with im x_(n+1) consists > > > > > > of > > > > > > exactly one element, which is the minimal element (in the standard > > > > > > ordering on Q+) in im x_n. > > > > > > > > Let X denote the subset of Q+ defined as follows: a positive > > > > > > rational > > > > > > number x is in X if and only if there exists some positive integer > > > > > > N > > > > > > such that, for all M > N, x is in the image of x_M. > > > > > > > > We are talking about X, right? > > > > > > > We are talking about a vase which is never emptied completely! > > > > > > > Hence it cannot be empty unless "infinity" is identical to "never". > > > > > But this describes potential infinity and excludes phantasies like > > > > > Cantor's finished diagonal number. > > > > > > > Regards, WM > > > > > > > Regards, WM > > > > > > The set X described above is certainly the empty set, as one can > > > > easily prove, despite im x_n being nonempty for each n. This seems to > > > > be a rigorous statement of what you are describing by talking about > > > > balls, vases, etc. I have never seen any mention of "potential > > > > infinity" or "completed infinity" which is precise enough to even be > > > > considered mathematics, > > > > > Completed infinity is the same as actual infinity. You need it to > > > obtain a diagonal number. Without that the number would never get > > > finished. > > > > > Potential infinity is the infinity of mathematics. It has always been > > > used until matheology started. > > > > I do not know what "matheology" is > > modern logic and set theory, a mixture between mathematics and > theology. The latter is prevailing. It is a area which does not exist outside of Wolkenmeukenheim, and does not include much of either logic or of set theory > > , nor what Wolkenmeukenheim are. Do these things > > actually have rigorous definitions? > > > That depends on your understanding of rigor. Those who understand and apply rigor have no need for any of Muekenheim's perverted notions of "completed infinity," "actual infinity," and "potential infinity" > > > > > > and talking about these ideas generally leads > > > >> to nothing better than confusion--it might be better to speak > > > > precisely about the limits or sets that you mean, in any given > > > > situation, rather than worrying whether they represent "potential > > > > infinity" or "completed infinity > > > > > You can talk about limits but you cannot talk about Cantor's diagonal > > > without actual infinity. If doing so, you get numbers that cannot be > > > named. This is the best proof showing that actual infinity is not to > > > be considered mathematics. > > > > > Regards, WM > > > > I do not know what "Cantor's diagonal" you are speaking of, but the > > method of proving that the real numbers are uncountable which is > > usually called "Cantor's diagonal argument" does not ever mention any > > "actual infinity." > > Nevertheless it requires actual infinity. It only requires a very simple an understanding of "infinite" which is still way beyond Muekenheim's capabilities. A non-empty set is finite if for any total ordering of its elements there is a last element, and is infinite otherwise, i.e., is totally orderable ssso that it does NOT have a last element. > > > > Perhaps you have some constructivist objections to mathematics as it > > is practiced using ZFC set theory and the nonconstructive proofs it > > allows. If that's so, then fine; Cantor's diagonal argument works fine > > in ZFC but a strict constructivist, who insists on a more restrictive > > set theory or a more restrictive logic, can find reasons to object to > > it.- > > A strict constructivist denies actaul infinity and, hence, uncountable > sets. But mathematics is not in thrall to strict constructionalists. > > Thus Cantor's method fails - here and everywhere. Only within the highly constraining boundaries of Wolkenmuekenheim. Those who are not so constrained, like Cantor, do not all fail.
From: Virgil on 27 Nov 2009 18:38 In article <ddb474bb-2113-4cff-a104-99f6ceed39ad(a)d21g2000yqn.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 27 Nov., 21:17, William Hughes <wpihug...(a)hotmail.com> wrote: > > On Nov 27, 3:33�pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > With only potential, i.e., not finished infinity, i.e., reasonable > > > infinity, �the diagonal number (exchanging 0 by 1) of the following > > > list can be found in the list as an entry: > > > > > 0.0 > > > 0.1 > > > 0.11 > > > 0.111 > > > ... > > > > Only in Wolkenmuekenheim where the argument goes > > > > � � �Every entry in the list has a fixed last 1 > > � � �The diagonal number does not have a fixed last 1 > There is not a fixed last entry. > > � � �[...] > Every diagonal number is in the list. > > Regards, WM Given any endless list of endless sequences from a two element set, such as Cantor's {m,w}, at which position in that list is the "anti-diagonal' to that list, Muekenheim? Unless Muekenheim has a specific and correct answer, he gets a failing grade.
From: WM on 28 Nov 2009 07:35 On 27 Nov., 22:42, William Hughes <wpihug...(a)hotmail.com> wrote: > On Nov 27, 5:24 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > > > On 27 Nov., 21:17, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > On Nov 27, 3:33 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > With only potential, i.e., not finished infinity, i.e., reasonable > > > > infinity, the diagonal number (exchanging 0 by 1) of the following > > > > list can be found in the list as an entry: > > > > > 0.0 > > > > 0.1 > > > > 0.11 > > > > 0.111 > > > > ... > > > > Only in Wolkenmuekenheim where the argument goes > > > > Every entry in the list has a fixed last 1 > > > The diagonal number does not have a fixed last 1 > > > There is not a fixed last entry > > So, every entry in the list has a fixed last 1. > (We don't need a fixed last entry to say this) > We still have > > Every entry in the list has a fixed last 1 > The diagonal number does not have a fixed last 1 > > > Every diagonal number is in the list. > > Only in Wolkenmuekenheim. Outside of Wolkenmuekenheim > there is only one diagonal number How do you know, unless you have seen the last? Regards, WM
From: William Hughes on 28 Nov 2009 08:11 On Nov 28, 8:35 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > On 27 Nov., 22:42, William Hughes <wpihug...(a)hotmail.com> wrote: > > > Every diagonal number is in the list. > > > Only in Wolkenmuekenheim. Outside of Wolkenmuekenheim > > there is only one diagonal number and it is not > > in the list. > > How do you know, unless you have seen the last? > You use induction to show that every entry in the list has a final 1. So you don't have to see an entry to know that it has a final 1 (there is no last entry in any case). There is a constructive proof that the diagonal number does not have a final 1. - William Hughes
From: Virgil on 28 Nov 2009 15:43
In article <f4e15df0-a3c0-48e4-959f-e341a9adf3ef(a)j4g2000yqe.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 27 Nov., 22:42, William Hughes <wpihug...(a)hotmail.com> wrote: > > On Nov 27, 5:24�pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > > > > > > > > > On 27 Nov., 21:17, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > > On Nov 27, 3:33�pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > > > With only potential, i.e., not finished infinity, i.e., reasonable > > > > > infinity, �the diagonal number (exchanging 0 by 1) of the following > > > > > list can be found in the list as an entry: > > > > > > > 0.0 > > > > > 0.1 > > > > > 0.11 > > > > > 0.111 > > > > > ... > > > > > > Only in Wolkenmuekenheim where the argument goes > > > > > > � � �Every entry in the list has a fixed last 1 > > > > � � �The diagonal number does not have a fixed last 1 > > > > > There is not a fixed last entry > > > > So, �every entry in the list has a fixed last 1. > > (We don't need a fixed last entry to say this) > > We still have > > > > � Every entry in the list has a fixed last 1 > > � The diagonal number does not have a fixed last 1 > > > > > Every diagonal number is in the list. > > > > Only in Wolkenmuekenheim. �Outside of Wolkenmuekenheim > > there is only one diagonal number > > How do you know, unless you have seen the last? > > Regards, WM Given a specific list of endless binary sequences, the so called Cantor diagonal is the result of a specific and unambiguous algorithm applied to that list, so it is, for any given list, unique, and not a member of the list from which it is constructed. Which WM would have known if he had any sense. |