Another AC anomaly?
in [Logic]
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From: WM on 27 Nov 2009 01:45 On 27 Nov., 02:50, William Hughes <wpihug...(a)hotmail.com> wrote: > On Nov 26, 4:51 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > On 26 Nov., 19:22, William Hughes <wpihug...(a)hotmail.com> wrote: > > > 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? > > Since outside of Wolkenmuekenheim there is no reason to > expect the number of balls to be continuous at infinity Why then do you expect the digits of Cantor's diagonal number to be "continuous" at infinity (contrary to being *not* at infinity)? Regards, WM
From: Virgil on 27 Nov 2009 02:41 In article <aa9e46c0-56da-4510-8345-8dee84745816(a)b2g2000yqi.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 27 Nov., 02:50, William Hughes <wpihug...(a)hotmail.com> wrote: > > On Nov 26, 4:51�pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > On 26 Nov., 19:22, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > 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? > > > > Since outside of Wolkenmuekenheim there is no reason to > > expect the number of balls to be continuous at infinity > > Why then do you expect the digits of Cantor's diagonal number to be > "continuous" at infinity (contrary to being *not* at infinity)? Why would anyone ever expect a numerical digit to be continuous? All the ones I am aware of are members of a finite set of discrete objects. And why would you expect to find a digit of any sort "at infinity", when there is no such a position as "at infinity".
From: Virgil on 27 Nov 2009 02:44 In article <9d0132ac-2c6b-447f-8515-22d5c69f1832(a)s20g2000yqd.googlegroups.com>, WM <mueckenh(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. Maybe in your muecked up worlds, but, fortunately, mathematics does not occur in such worlds.
From: WM on 27 Nov 2009 04:30 On 27 Nov., 08:41, Virgil <Vir...(a)home.esc> wrote: > In article > <aa9e46c0-56da-4510-8345-8dee84745...(a)b2g2000yqi.googlegroups.com>, > > > > > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > On 27 Nov., 02:50, William Hughes <wpihug...(a)hotmail.com> wrote: > > > On Nov 26, 4:51 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > On 26 Nov., 19:22, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > 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? > > > > Since outside of Wolkenmuekenheim there is no reason to > > > expect the number of balls to be continuous at infinity > > > Why then do you expect the digits of Cantor's diagonal number to be > > "continuous" at infinity (contrary to being *not* at infinity)? > > Why would anyone ever expect a numerical digit to be continuous? > > All the ones I am aware of are members of a finite set of discrete > objects. And there is none that does not belong to a rational number. > > And why would you expect to find a digit of any sort "at infinity", when > there is no such a position as "at infinity". If there is no "at infinity", then there cannot be a "behind infinity", so there is no omega and no omega + 1. In fact you are right - as so often. There is no "at infinity". The vase is never empty. There is no smallest positive rational, There are not all rationals. Regards, WM
From: Alan Smaill on 27 Nov 2009 05:43
WM <mueckenh(a)rz.fh-augsburg.de> writes: > 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. But you lose control at infinity! So your "hence" doesn't work. I have that on good authority. > Regards, WM > > Regards, WM -- Alan Smaill |