Another AC anomaly?
in [Logic]
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From: WM on 27 Nov 2009 14:23 On 27 Nov., 18:50, William Hughes <wpihug...(a)hotmail.com> wrote: > On Nov 27, 1:32 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > So all steps have been done whereas the last is pending. > > Nope. Outside of Wolkenmeukenheim > we can do all steps without doing a last step So all steps > have been done, however there is no last step so the last > is not pending. Whatever you may think. If you wish to argue that the sequence of balls has limt 0 then you are outside of mathematics. And there is no need to further answer your "arguments". Regards, WM
From: WM on 27 Nov 2009 14:33 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. , nor what "completed infinity," > "actual infinity," and "potential infinity" are. Do these things > actually have rigorous definitions? > That depends on your understanding of rigor. > > > > 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. > > 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. 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 .... Thus Cantor's method fails - here and everywhere. Regards, WM
From: WM on 27 Nov 2009 14:34 On 27 Nov., 20:11, Virgil <Vir...(a)home.esc> wrote: > In article > <ac583807-eb9d-4ae4-86f2-37f329692...(a)k17g2000yqh.googlegroups.com>, > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > Therefore the vase is never empty - and the only error is your > > assertion it would be empty after the last step and there was a state > > of the vase "at infinity". > > While there is no 'last step' in this infinite sequence of steps, there > is an 'after all steps' because there is a time at which every step has > been completed. Whatever you may think. If you wish to argue that the sequence of balls has limit 0 then you are outside of mathematics. And there is no need to further answer your "arguments". Regards, WM
From: Virgil on 27 Nov 2009 14:38 In article <a02569c0-3b73-4c60-99f3-fbb1e9d28391(a)b15g2000yqd.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 27 Nov., 18:50, William Hughes <wpihug...(a)hotmail.com> wrote: > > On Nov 27, 1:32�pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > So all steps have been done whereas the last is pending. > > > > Nope. �Outside of Wolkenmeukenheim > > we can do all steps without doing a last step �So all steps > > have been done, �however there is no last step so the last > > is not pending. > > Whatever you may think. If you wish to argue that the sequence of > balls has limt 0 then you are outside of mathematics. And there is no > need to further answer your "arguments". Since Mueckenheim so seldom manages to get himself inside of mathematics, he is hardly an authority on what goes on there. N mathematics, it is quite possible, even common, for an infinite sequence to be completed and have a limit which is not a member of the sequence itself. And if the problem of the balls and urn can be stated unambiguously, and it has been, then its analysis can be done inside of mathematics, even though it does not appear to be possible in Wolkenmeukenheim.
From: William Hughes on 27 Nov 2009 14:39
On Nov 27, 3:23 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > On 27 Nov., 18:50, William Hughes <wpihug...(a)hotmail.com> wrote: > > > On Nov 27, 1:32 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > So all steps have been done whereas the last is pending. > > > Nope. Outside of Wolkenmeukenheim > > we can do all steps without doing a last step So all steps > > have been done, however there is no last step so the last > > is not pending. > > Whatever you may think <snip unsupported assertion> An unsupported assertion by WM will work in Wolkenmuekenheim, but is of no use outside Wolkenmeukenheim. - William Hughes |