Another AC anomaly?
in [Logic]
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From: William Hughes on 27 Nov 2009 08:48 On Nov 27, 2:45 am, 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 > <snip attempted change of subject> The argument why the change is the number of balls is not problem is simple. Outside Wolkenmuekenheim There is a contradiction at the last step There is no last step There is no contradiction. Of course inside Wolkenmuekenheim ANYTHING can change (lists, sets, last elements, fixed last elements etc.) There is a contradiction at the last step. There is a last step (it is something that changes) There is a contradiction. - William Hughes
From: WM on 27 Nov 2009 09:25 On 27 Nov., 11:43, Alan Smaill <sma...(a)SPAMinf.ed.ac.uk> wrote: > WM <mueck...(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 does Cantor. > So your "hence" doesn't work. It works if there is anyone who does not lose control at infinity. That's enough. Regards, WM
From: WM on 27 Nov 2009 09:29 On 27 Nov., 14:48, William Hughes <wpihug...(a)hotmail.com> wrote: ********************* 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 ***************** > > The argument why the change is the number > of balls is not problem is simple. Outside Wolkenmuekenheim > > There is a contradiction at the last step > There is no last step > There is no contradiction. 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". Regards, WM
From: Alan Smaill on 27 Nov 2009 09:31 WM <mueckenh(a)rz.fh-augsburg.de> writes: > On 27 Nov., 11:43, Alan Smaill <sma...(a)SPAMinf.ed.ac.uk> wrote: > > WM <mueck...(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 does Cantor. Good to know you agree that *you* lose control. Let's leave Cantor out of it. > > So your "hence" doesn't work. > > It works if there is anyone who does not lose control at infinity. > That's enough. If you are not in control, it doesn't work for you; you can't say anything about what does or doesn't happen at infinity, except by pretending that it must be the same as the finite case. That pretence is worthless when you admit you have lost control. > Regards, WM -- Alan Smaill
From: A on 27 Nov 2009 10:21
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, 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." |