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From: Aatu Koskensilta on 1 Dec 2009 23:14 Virgil <Virgil(a)home.esc> writes: > There are classes in NBG which are not sets and which do not exist in > ZFC, ergo, they are not "equivalent". NBG is essentially just a notational variant of ZFC. The differences between these two theories are of technical logical interest only, and don't reflect anything in their mathematical content. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Dik T. Winter on 2 Dec 2009 08:14 In article <12b419b5-1a9f-42c1-b9bc-0e8a2cce2da8(a)z41g2000yqz.googlegroups.com> WM <mueckenh(a)rz.fh-augsburg.de> writes: > On 1 Dez., 14:57, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: .... > > > You may write this as often as you like, but you are wrong. If there > > > is a limit set then there is a limit cardinality, namely the number of > > > elements in that limit set. Everything else is nonsense. > > > > Can you prove the assertion that the limit of the cardinalities is the > > cardinality of the limit? > > With pleasure. My assertion is obvious if the limit set actually > exists. No it is not. > Limit cardinality = Cardinality of the limit-set. You are conflating within limit cardinality the limit of the cardinalities and the cardinality of the limit. You have to prove they are equal. > If it does not exist, set theory claiming the existence of actual > infinity is wrong. Both can exist but they are not necessarily equal, you have to *prove* that. > > I can prove that it can be false. As I wrote, > > given: > > S_n = {n, n+1} > > we have (by the definition of limit of sets: > > lim{n -> oo} S_n = {} > > This is wrong. You have never looked at the definition I think. Given a sequence of sets S_n then: lim sup{n -> oo} S_n contains those elements that occur in infinitely many S_n lim inf{n -> oo} S_n contains those elements that occur in all S_n from a certain S_n (which can be different for each element). lim{n -> oo} S_n exists whenever lim sup and lim inf are equal. With this definition lim{n -> oo} S_n exists and is equal to {}. > ong. You may find it helpful to see the approach where I > show a related example to my students. > > http://www.hs-augsburg.de/~mueckenh/GU/GU12.PPT#394,22,Folie 22 I see nothing related there. It just shows (I think) an open cube and a cylinder. > > and so > > 2 = lim{n -> oo} | S(n) | != | lim{n -> oo} S_n | = 0 > > or can you show what I wrote is wrong? > > Yes I can.* If actual infinity exists*, then the limit set exists. I have still no idea what the mathematical definition of "actual infinity" is, but given the definitions above, the limit of the sequence of sets exists and is the empty set. > Then the limit set has a cardinal number which can be determined > simply by counting its elements. > A simple example is > | lim[n --> oo] {1} | = | {1} | = lim[n --> oo] |{1}| = 1. Right. And as in the above example the limit is the empty set, we can count the elements and come at 0. > If you were right, that | lim[n --> oo] S_n | =/= lim[n --> oo] |S_n| > was possible, then the limit set would not actually exist (such that > its elements could be counted and a different limit could be shown > wrong). The limit set is empty. > > If so, what line is wrong? > > Wrong is your definition of limit set, or set theory claiming actual > infinity as substantially existing, or both. What is *your* definition of the limit of a sequence of sets? -- 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 2 Dec 2009 09:34 On 2 Dez., 14:14, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <12b419b5-1a9f-42c1-b9bc-0e8a2cce2...(a)z41g2000yqz.googlegroups..com> WM <mueck...(a)rz.fh-augsburg.de> writes: > > On 1 Dez., 14:57, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > ... > > > > You may write this as often as you like, but you are wrong. If there > > > > is a limit set then there is a limit cardinality, namely the number of > > > > elements in that limit set. Everything else is nonsense. > > > > > > Can you prove the assertion that the limit of the cardinalities is the > > > cardinality of the limit? > > > > With pleasure. My assertion is obvious if the limit set actually > > exists. > > No it is not. > > > Limit cardinality = Cardinality of the limit-set. > > You are conflating within limit cardinality the limit of the cardinalities > and the cardinality of the limit. You have to prove they are equal. If the limit set exists, then it has a cardinal number, hasn't it? > > > If it does not exist, set theory claiming the existence of actual > > infinity is wrong. > > Both can exist but they are not necessarily equal, you have to *prove* that. If a set exists (and if ZFC is correct), then that set has a cardinality. If the limit set exists, then it has a cardinality. I call that the cardinality of the limit set, abbreviated by limit cardinality. If the limit of cardinalities differs, then either the calculation is wrong or the theory whereupon the calculation is based. It is so simple. There is as little proof required as in necessary to show that you are Dik T. Winter. If there is any theory showing that you are not DTW or that you are 20 meters tall, then that theory is simply wrong - without further proof. > > > > I can prove that it can be false. As I wrote, > > > given: > > > S_n = {n, n+1} > > > we have (by the definition of limit of sets: > > > lim{n -> oo} S_n = {} > > > > This is wrong. > > You have never looked at the definition I think. Given a sequence of sets > S_n then: > lim sup{n -> oo} S_n contains those elements that occur in infinitely > many S_n > lim inf{n -> oo} S_n contains those elements that occur in all S_n from > a certain S_n (which can be different for each element). > lim{n -> oo} S_n exists whenever lim sup and lim inf are equal. > With this definition lim{n -> oo} S_n exists and is equal to {}. Then the theory is wrong. If another set exists and the calculation of its cardinality gives not the cardinality of that set, what would you conclude? Well, in set the theory the limit set is claimed to exist. > > > ong. You may find it helpful to see the approach where I > > show a related example to my students. > > > >http://www.hs-augsburg.de/~mueckenh/GU/GU12.PPT#394,22,Folie22 > > I see nothing related there. It just shows (I think) an open cube and a > cylinder. You must move on. There is always a number remaining in the cylinder (the example stretches only until 6, but you should imagine how it continues). It shows: If you union all natural numbers within the cube, then it is said that you get all natural numbers, the complete set N. But if you union all natural numbers such that each one makes an intermediate stop in the cylinder and does not move on before the next one, n+1, has dropped in, then you cannot union all natural numbers within the cube. This simple example shows, that it is impossible to union all natural numbers at all. Students easily understand, that this union is but a silly idea of unmathematical people. > > > > and so > > > 2 = lim{n -> oo} | S(n) | != | lim{n -> oo} S_n | = 0 > > > or can you show what I wrote is wrong? > > > > Yes I can.* If actual infinity exists*, then the limit set exists. > > I have still no idea what the mathematical definition of "actual infinity" > is, but given the definitions above, the limit of the sequence of sets > exists and is the empty set. Actual infinity is completed infinity. Unless actual infinity is assumed, no infinite counting comes to an end. No diagonal number comes ever into being. > > > Then the limit set has a cardinal number which can be determined > > simply by counting its elements. > > A simple example is > > | lim[n --> oo] {1} | = | {1} | = lim[n --> oo] |{1}| = 1. > > Right. And as in the above example the limit is the empty set, we can count > the elements and come at 0. This proves set theory wrong. In the vase (and in the cylinder in http://www.hs-augsburg.de/~mueckenh/GU/GU12.PPT#394,22,Folie 22 there is always at least one element. Hence cardinality is 1 in the limit. > > > If you were right, that | lim[n --> oo] S_n | =/= lim[n --> oo] |S_n| > > was possible, then the limit set would not actually exist (such that > > its elements could be counted and a different limit could be shown > > wrong). > > The limit set is empty. The limit set is the same as in case 1, 1, 1, ... --> 1 It is never empty and has never cardinality 0. > > > > If so, what line is wrong? > > > > Wrong is your definition of limit set, or set theory claiming actual > > infinity as substantially existing, or both. > > What is *your* definition of the limit of a sequence of sets? There is no actual infinity. Hence there is no limit set. The only thing you can do is this: You can look into my cylinder at any time you like and you will find at least one element inside. Please note the difference: The limit of the *sequence* (1/n) is 0. But there is no term 1/n = 0. The limit is not assumed by the sequence. It is defined by means of epsilon. The limit of ({1, 2, 3, ...n}) however is assumed to exist as N (and my cylinder being empty). Our discussion shows that this assumption is untenable. Regards, WM
From: Dik T. Winter on 2 Dec 2009 10:27 In article <a7ae8268-3626-4a9f-9902-65662d39d7a3(a)m25g2000yqc.googlegroups.com> WM <mueckenh(a)rz.fh-augsburg.de> writes: > On 2 Dez., 14:14, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: .... > > You are conflating within limit cardinality the limit of the cardinalities > > and the cardinality of the limit. You have to prove they are equal. > > If the limit set exists, then it has a cardinal number, hasn't it? Right, I never denied that. > > > If it does not exist, set theory claiming the existence of actual > > > infinity is wrong. > > > > Both can exist but they are not necessarily equal, you have to *prove* > > that. > > If a set exists (and if ZFC is correct), then that set has a > cardinality. If the limit set exists, then it has a cardinality. I > call that the cardinality of the limit set, abbreviated by limit > cardinality. Limit cardinality is confusing because it can either mean the cardinality of the limit set or the limit of the cardinalities. > 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? > It is so simple. There is as little proof required as in necessary to > show that you are Dik T. Winter. Pray show that I am Dik T. Winter. > > > > You have never looked at the definition I think. Given a sequence of sets > > S_n then: > > lim sup{n -> oo} S_n contains those elements that occur in infinitely > > many S_n > > lim inf{n -> oo} S_n contains those elements that occur in all S_n from > > a certain S_n (which can be different for each > > element). > > lim{n -> oo} S_n exists whenever lim sup and lim inf are equal. > > With this definition lim{n -> oo} S_n exists and is equal to {}. > > Then the theory is wrong. What part of the theory is wrong? > If another set exists and the calculation of > its cardinality gives not the cardinality of that set, what would you > conclude? There are not two different sets. What are you babbling about? > Well, in set the theory the limit set is claimed to exist. Right. See above for the definition. > > >http://www.hs-augsburg.de/~mueckenh/GU/GU12.PPT#394,22,Folie22 > > > > I see nothing related there. It just shows (I think) an open cube and a > > cylinder. > > You must move on. There is always a number remaining in the cylinder > (the example stretches only until 6, but you should imagine how it > continues). I have no idea what you are talking about. I see only a picture of a cylinder with a few numbers on it and an open cube. > It shows: If you union all natural numbers within the cube, then it is > said that you get all natural numbers, the complete set N. But if you > union all natural numbers such that each one makes an intermediate > stop in the cylinder and does not move on before the next one, n+1, > has dropped in, then you cannot union all natural numbers within the > cube. This simple example shows, that it is impossible to union all > natural numbers at all. Students easily understand, that this union is > but a silly idea of unmathematical people. I do not understand this at all. > > > > and so > > > > 2 = lim{n -> oo} | S(n) | != | lim{n -> oo} S_n | = 0 > > > > or can you show what I wrote is wrong? > > > > > > Yes I can.* If actual infinity exists*, then the limit set exists. > > > > I have still no idea what the mathematical definition of "actual > > infinity" is, but given the definitions above, the limit of the > > sequence of sets exists and is the empty set. > > Actual infinity is completed infinity. I have no idea what the mathematical definition of "completed infinity" is. > Unless actual infinity is assumed, no infinite counting comes to an > end. No diagonal number comes ever into being. As I have no idea about what "actual infinity" is, this makes no sense to me. > > > Then the limit set has a cardinal number which can be determined > > > simply by counting its elements. > > > A simple example is > > > | lim[n --> oo] {1} | = | {1} | = lim[n --> oo] |{1}| = 1. > > > > Right. And as in the above example the limit is the empty set, we can > > count the elements and come at 0. > > This proves set theory wrong. In the vase (and in the cylinder in > http://www.hs-augsburg.de/~mueckenh/GU/GU12.PPT#394,22,Folie 22 > there is always at least one element. Hence cardinality is 1 in the > limit. No, that is the limit of the cardinalities. > > > If you were right, that | lim[n --> oo] S_n | =/= lim[n --> oo] |S_n| > > > was possible, then the limit set would not actually exist (such that > > > its elements could be counted and a different limit could be shown > > > wrong). > > > > The limit set is empty. > > The limit set is the same as in case 1, 1, 1, ... --> 1 > It is never empty and has never cardinality 0. No, you are confusing limits of sets and limits of numbers. The case {1}, {1}, {1}, {1} --> {1} is different from {1}, {2}, {3}, {4} --> ? > > > > If so, what line is wrong? > > > > > > Wrong is your definition of limit set, or set theory claiming actual > > > infinity as substantially existing, or both. > > > > What is *your* definition of the limit of a sequence of sets? > > There is no actual infinity. Hence there is no limit set. The only > thing you can do is this: You can look into my cylinder at any time > you like and you will find at least one element inside. > > Please note the difference: The limit of the *sequence* (1/n) is 0. > But there is no term 1/n = 0. The limit is not assumed by the > sequence. It is defined by means of epsilon. Right, and the limit of a sequence of sets is a set, but that limit is not assumed by the sequence. It is defined by ohter means (see higher up). > The limit of ({1, 2, 3, ...n}) however is assumed to exist as N (and > my cylinder being empty). Our discussion shows that this assumption is > untenable. But the limit is not assumed by the sequence. I do not understand why you do allow the limit of the sequence (1/n) to be 0 but not the limit of the sequence {n} be {}. -- 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 2 Dec 2009 11:16
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. If both differ, then there is something fishy about the theory. > > > 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: 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. Same holds for cardinals of sets. You form a set by unioning its elements, resulting in the complete set. 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. The set N can be constructed by unioning all naturals. Or the set can be taken from the shelf. Both ways must lead to the same result. How could it be that the due cardinalities differ? 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. Regards, WM I think that this answers all the questions contained in the following. Therefore I extinguish it. |