Another AC anomaly?
in [Logic]
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From: Dik T. Winter on 3 Dec 2009 07:52 In article <0c635f53-f5c4-4320-8825-de05f021a428(a)m3g2000yqf.googlegroups.com> WM <mueckenh(a)rz.fh-augsburg.de> writes: > On 2 Dez., 16:27, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > Limit cardinality is confusing because it can either mean the cardinality > > of the limit set or the limit of the cardinalities. > > If the limit set is "assumed" by the sequence of sets, if for instance > N is given by unioning all naturla numbers, then tghe limit set has to > have the limit cardinality. The limit set is not necessarily "assumed" by the sequence of sets (if by that you mean that there is a set in the sequence that is equal to the limit set, if you mean something else I do not understand it at all). When you mean with your statement about N: N = union{n is natural} {n} then that is not a limit. Check the definitions about it. So what you wrote is complete nonsense. > If both differ, then there is something > fishy about the theory. There is something fishy about how you think things work in mathematics. > > > If the limit of cardinalities differs, then either the > > > calculation is wrong or the theory whereupon the calculation is based. > > > > Why? I have given you a precise definition of the limit of a sequence of > > sets. With that definition the limit of cardinalities is different from > > the cardinality of the limit, as is easily calculated. So, what part of > > the theory is wrong? > > My argument is comparable to the following: So below is your argument: > If you want to prove that the determinant of a matrix M with not > ecclusively linear independent rows is det(M) = 0, you go two ways: > You multiply a row of the matrix M by 0, and you empty a row of M by > elementary operations which do not change the determinant. Then you > have proved that the original matrix M has the determinant 0. So you argue that each matrix has determinant 0? If not what do you mean with that? > Same holds for cardinals of sets. You form a set by unioning its > elements, resulting in the complete set. You can not unite the elements if the elements are not sets in themselves and when the elements are sets, the union of the elements is different from the original set. Consider: A = {{a, b}, {b, c}, {c, d}} uniting the elements gives: B = {a, b, c, d} quite different. > This cannot change the > connection between cardinal number and set during the whole process of > formation. For every step the cardinal number and the number of > elements of the set are equal. What do you mean with "every step"? Uniting sets is a single step operation. > Same holds for the cylinder. Its contents is > {1}, {1}, {1}, {1} , ... > and that is not different from > {1}, {2}, {3}, {4} , ... > as you can see by renaming the elements. I do not understand this at all. > I think that this answers all the questions contained in the > following. Therefore I extinguish it. No. I asked you for a mathematical definition of "actual infinity" and you told me that it was "completed infinity". Next I asked you for a mathematical definition of "completed infinity" but you have not given an answer. So I still do not know what either "actual infinity" or "completed infinity" are. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: WM on 3 Dec 2009 09:46 On 3 Dez., 13:52, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <0c635f53-f5c4-4320-8825-de05f021a...(a)m3g2000yqf.googlegroups.com> WM <mueck...(a)rz.fh-augsburg.de> writes: > > On 2 Dez., 16:27, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > Limit cardinality is confusing because it can either mean the cardinality > > > of the limit set or the limit of the cardinalities. > > > > If the limit set is "assumed" by the sequence of sets, if for instance > > N is given by unioning all naturla numbers, then tghe limit set has to > > have the limit cardinality. > > The limit set is not necessarily "assumed" by the sequence of sets (if by > that you mean that there is a set in the sequence that is equal to the > limit set, if you mean something else I do not understand it at all). When > you mean with your statement about N: > N = union{n is natural} {n} > then that is not a limit. Check the definitions about it. It is a limit. That is independent from any definition. > > So below is your argument: > > > If you want to prove that the determinant of a matrix M with not > > ecclusively linear independent rows is det(M) = 0, you go two ways: > > You multiply a row of the matrix M by 0, and you empty a row of M by > > elementary operations which do not change the determinant. Then you > > have proved that the original matrix M has the determinant 0. > > So you argue that each matrix has determinant 0? If not what do you > mean with that? > > > Same holds for cardinals of sets. You form a set by unioning its > > elements, resulting in the complete set. > > You can not unite the elements if the elements are not sets in themselves > and when the elements are sets, the union of the elements is different > from the original set. Consider: > A = {{a, b}, {b, c}, {c, d}} > uniting the elements gives: > B = {a, b, c, d} > quite different. Not at all different with respect to being a limit process. > > > This cannot change the > > connection between cardinal number and set during the whole process of > > formation. For every step the cardinal number and the number of > > elements of the set are equal. > > What do you mean with "every step"? Uniting sets is a single step > operation. > No. Uniting two sets or singlets or elements is a single-step operation. Uniting infinitely many sets or singlets or elements is a limit process. > > Same holds for the cylinder. Its contents is > > {1}, {1}, {1}, {1} , ... > > and that is not different from > > {1}, {2}, {3}, {4} , ... > > as you can see by renaming the elements. > > I do not understand this at all. I see. You seem to share that fate with most so-called matematicians. > > > I think that this answers all the questions contained in the > > following. Therefore I extinguish it. > > No. I asked you for a mathematical definition of "actual infinity" and you > told me that it was "completed infinity". Next I asked you for a mathematical > definition of "completed infinity" but you have not given an answer. So I > still do not know what either "actual infinity" or "completed infinity" are. Both are nonsense. But both are asumed to make sense in set theory. The axiom of infinity is adefinition of actual infinity. "There *exists* a set such that ..." Without that axiom there is only potential infinity, namely Peano arithmetic. Regards, WM Regards, WM
From: Dik T. Winter on 3 Dec 2009 10:27 In article <27aee2d6-5966-4b62-a026-fea13e0bad6c(a)h2g2000vbd.googlegroups.com> WM <mueckenh(a)rz.fh-augsburg.de> writes: > On 3 Dez., 13:52, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: .... > > > If the limit set is "assumed" by the sequence of sets, if for instance > > > N is given by unioning all naturla numbers, then tghe limit set has to > > > have the limit cardinality. > > > > The limit set is not necessarily "assumed" by the sequence of sets (if by > > that you mean that there is a set in the sequence that is equal to the > > limit set, if you mean something else I do not understand it at all). > > When you mean with your statement about N: > > N = union{n is natural} {n} > > then that is not a limit. Check the definitions about it. > > It is a limit. That is independent from any definition. It is not a limit. Nowhere in the definition of that union a limit is used or mentioned. > > You can not unite the elements if the elements are not sets in themselves > > and when the elements are sets, the union of the elements is different > > from the original set. Consider: > > A = {{a, b}, {b, c}, {c, d}} > > uniting the elements gives: > > B = {a, b, c, d} > > quite different. > > Not at all different with respect to being a limit process. What limit process? Are there cases that A and B are equal? > > > This cannot change the > > > connection between cardinal number and set during the whole process of > > > formation. For every step the cardinal number and the number of > > > elements of the set are equal. > > > > What do you mean with "every step"? Uniting sets is a single step > > operation. > > No. Uniting two sets or singlets or elements is a single-step > operation. > Uniting infinitely many sets or singlets or elements is a limit > process. You are wrong. We have had this discussion before. Uniting a collection of sets is a single step process, regardless the number of sets involved (which number can be uncountable). Look up the definition of the union of a collection of sets in ZF. When you do not understand how some mathematics work, do not critisise it. > > > Same holds for the cylinder. Its contents is > > > {1}, {1}, {1}, {1} , ... > > > and that is not different from > > > {1}, {2}, {3}, {4} , ... > > > as you can see by renaming the elements. > > > > I do not understand this at all. > > I see. You seem to share that fate with most so-called matematicians. The only thing I see about that cylinder is that it contains some numbers, never changing. > > > I think that this answers all the questions contained in the > > > following. Therefore I extinguish it. > > > > No. I asked you for a mathematical definition of "actual infinity" and > > you told me that it was "completed infinity". Next I asked you for a > > mathematical definition of "completed infinity" but you have not given > > an answer. So I still do not know what either "actual infinity" or > > "completed infinity" are. > > Both are nonsense. But both are asumed to make sense in set theory. No, set theory does not contain a definition of either of them. > The axiom of infinity is adefinition of actual infinity. > "There *exists* a set such that ..." > Without that axiom there is only potential infinity, namely Peano > arithmetic. I see neither a definition of the words "actual infinity" neither a definition of "potential infinity". Or do you mean that "potential infinity" is Peano arithmetic (your words seem to imply that)? So we can say that in "potential infinity" consists of a set of axioms? -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Virgil on 3 Dec 2009 15:21 In article <27aee2d6-5966-4b62-a026-fea13e0bad6c(a)h2g2000vbd.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 3 Dez., 13:52, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article > > <0c635f53-f5c4-4320-8825-de05f021a...(a)m3g2000yqf.googlegroups.com> WM > > <mueck...(a)rz.fh-augsburg.de> writes: > > �> On 2 Dez., 16:27, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > �> > Limit cardinality is confusing because it can either mean the > > cardinality > > �> > of the limit set or the limit of the cardinalities. > > �> > > �> If the limit set is "assumed" by the sequence of sets, if for instance > > �> N is given by unioning all naturla numbers, then tghe limit set has to > > �> have the limit cardinality. > > > > The limit set is not necessarily "assumed" by the sequence of sets (if by > > that you mean that there is a set in the sequence that is equal to the > > limit set, if you mean something else I do not understand it at all). �When > > you mean with your statement about N: > > � � �N = union{n is natural} {n} > > then that is not a limit. �Check the definitions about it. > > It is a limit. That is independent from any definition. Every mathemtical "limit" notion has to be defined, as there is no ur-limit notion which is universal in mathematics. Of course, what goes on in Wolkenmuekenheim is largely irrelevant to mathematics. > > So below is your argument: > > > > �> If you want to prove that the determinant of a matrix M with not > > �> ecclusively linear independent rows is det(M) = 0, you go two ways: > > �> You multiply a row of the matrix M by 0, and you empty a row of M by > > �> elementary operations which do not change the determinant. Then you > > �> have proved that the original matrix M has the determinant 0. > > > > So you argue that each matrix has determinant 0? �If not what do you > > mean with that? > > > > �> Same holds for cardinals of sets. You form a set by unioning its > > �> elements, resulting in the complete set. > > > > You can not unite the elements if the elements are not sets in themselves > > and when the elements are sets, the union of the elements is different > > from the original set. �Consider: > > � �A = {{a, b}, {b, c}, {c, d}} > > uniting the elements gives: > > � �B = {a, b, c, d} > > quite different. > > Not at all different with respect to being a limit process. Quite different in mathematics, at least outside of Wolkenmuekenheim. > > > > �> � � � � � � � � � � � � � � � � � � � � �This cannot change the > > �> connection between cardinal number and set during the whole process of > > �> formation. For every step the cardinal number and the number of > > �> elements of the set are equal. > > > > What do you mean with "every step"? �Uniting sets is a single step > > operation. > > > > No. Uniting two sets or singlets or elements is a single-step > operation. > Uniting infinitely many sets or singlets or elements is a limit > process. Not when using the Peano axioms. > > > �> Same holds for the cylinder. Its contents is > > �> � {1}, {1}, {1}, {1} , ... > > �> and that is not different from > > �> � {1}, {2}, {3}, {4} , ... > > �> as you can see by renaming the elements. > > > > I do not understand this at all. > > I see. You seem to share that fate with most so-called matematicians. > > > > �> I think that this answers all the questions contained in the > > �> following. Therefore I extinguish it. > > > > No. �I asked you for a mathematical definition of "actual infinity" and you > > told me that it was "completed infinity". �Next I asked you for a > > mathematical > > definition of "completed infinity" but you have not given an answer. �So I > > still do not know what either "actual infinity" or "completed infinity" > > are. > > Both are nonsense. Then don't base your arguments on them. And neither is needed in any version of set theory outside of Wolkenmuekenheim. > The axiom of infinity is adefinition of actual infinity. > "There *exists* a set such that ..." > Without that axiom there is only potential infinity, namely Peano > arithmetic. Without that axiom, or something like it, there is no Peano arithmetic.
From: WM on 3 Dec 2009 16:11
On 3 Dez., 16:27, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <27aee2d6-5966-4b62-a026-fea13e0ba...(a)h2g2000vbd.googlegroups.com> WM <mueck...(a)rz.fh-augsburg.de> writes: > > On 3 Dez., 13:52, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > ... > > > > If the limit set is "assumed" by the sequence of sets, if for instance > > > > N is given by unioning all naturla numbers, then tghe limit set has to > > > > have the limit cardinality. > > > > > > The limit set is not necessarily "assumed" by the sequence of sets (if by > > > that you mean that there is a set in the sequence that is equal to the > > > limit set, if you mean something else I do not understand it at all). > > > When you mean with your statement about N: > > > N = union{n is natural} {n} > > > then that is not a limit. Check the definitions about it. > > > > It is a limit. That is independent from any definition. > > It is not a limit. Nowhere in the definition of that union a limit is used > or mentioned. 1) N is a set that follows (as omega, but that is not important) from the axiom of infinity. You can take it "from the shelf". 2) N is the limit of the sequence a_n = ({1, 2, 3, ...,n}) 3) N is the limit, i.,e. the infinite union of singletons {1} U {2} U ... This is fact. But if (3) is correct, then N must also be the limit of the process described in my http://www.hs-augsburg.de/~mueckenh/GU/GU12.PPT#394,22,Folie 22 without and *with* the intermediate cylinder. Then the cylinder must be empty in the limit, and cardinality of the limit set in the cylinder must be 0. That, however, is wrong, simply because the cylinder is never empty. Regards, WM |