Another AC anomaly?
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From: WM on 27 Nov 2009 12:36 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. > 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
From: WM on 27 Nov 2009 12:40 On 27 Nov., 17:02, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <ac583807-eb9d-4ae4-86f2-37f329692...(a)k17g2000yqh.googlegroups..com> WM <mueck...(a)rz.fh-augsburg.de> writes: > ... > > Therefore the vase is never empty - > > Similar in the sequence 1/n the elements value is never zero. Correct. But in the sequence 1,1,1... the limit is 1 with no doubt. > You may verify that in the case you proposed lim X_n does exist and is equal > to the empty set. > > Note: > 1 = lim |X_n| != |lim X_n| = 0 > but that should not come as a surprise. Let one ball rest in the vase in eternity. The limit will be 1. Consider the sequence 1, 101, 1, 101, 1, 101, ... There is no limit. Nevertheless the sequence of minima 1,1,1,... has limit 1 as before. Therefore the result of set theory shows that set theory is not mathematics. Regards, WM
From: William Hughes on 27 Nov 2009 12:50 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. - William Hughes
From: A on 27 Nov 2009 12:55 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, nor what "completed infinity," "actual infinity," and "potential infinity" are. Do these things actually have rigorous definitions? > > > 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." 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.
From: Virgil on 27 Nov 2009 14:03
In article <ee22fd89-2e1f-464d-a456-0049caa7d693(a)j19g2000yqk.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > 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. There are several that do not belong to 1/3. > > > > 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 the set of rationals, for example, there can be "before" and an "after" without there being an "at", e.g. before and after sqrt(2). So your claim requires proofs whch tou do not have. > > In fact you are right - as so often. There is no "at infinity". The > vase is never empty. The case starts empty, so WM is again trivially wrong. > There is no smallest positive rational, There are > not all rationals. Which ones are not there? |