An uncountable countable set
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From: mueckenh on 16 Jul 2006 04:42 Virgil schrieb: > > > > > > Each ordinal has a successor, even limit ordinals like N, and any > > > ordinal can be indexed by the members of its successor. > > > > Each ordinal can be indexed by itself. 1 is indexed by 1. 1,2 is > > indexed by 1 and 2. 1,2,3 is indexed by 1,2 and 3 and so on. But all > > naturals are not indexed by all naturals? Extremely ridiculous! > > Extremely! > Every ordinal has successors, whether in ZF or NBG or most other set > theories, and to index any of them it is sufficient to use at the > members of any successor. But the successor of 4 does not consist of the ingredients of 4 only. So the successor of all natural numbers cannot consist of only all natural numbers only. This is obvious and therefore Tony Orlow is one of the few persons who realize he truth and you and many others are not among them. Regards, WM
From: mueckenh on 16 Jul 2006 05:02 Dik T. Winter schrieb: > In article <1152979000.961294.290560(a)m79g2000cwm.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > In article <1152884145.033413.104250(a)75g2000cwc.googlegroups.com> muecken= > > h(a)rz.fh-augsburg.de writes: > ... > Let's start with an English translation, makes it quite a bit easier for > me (I have not to do two translations with three different languages): > Works, p. 214, translated: > The question, through which transformations the ordinal number of a > well-ordered set is changed, and through which it is not changed, is > easily answered, those, and only those transformations leave the > ordinal number unchanged that can be rewritten as a finite or infinite > set of transpositions, that is, of interchanges of two elements. > Agree that this is a reasonable translation? OK > > > > No, but still, not *any* set of transpositions is permitted. Only those > > > sets of transpositions that are the result of rewriting a transformation > > > ("Umformungen"). So it remains interesting what Cantor is meaning when > > > he writes "Umformungen". And when he is meaning with that word arbitrary > > > re-orderings, he is wrong (because *all* re-orderings can be rewritten as > > > an infinite set of transpositions). > > > > You do completely misunderstand what is defined and what is the > > defintion. > > I think not. > > > Defined is Umformungen: "diejenigen und nur diejenigen Umformungen = > > those and only those transformations" > > Yes, so that is not a definition of transformations, it restricts the > transformations to a subset of them, namely only those transformations > that have a particular property. Of course. "Umformungen" includes many transformations like those of iron to hammers. > Apparently there are transformations > that do *not* have that property. So the question remains, what is the > *definition* of "transformations"? (I can easily enough give a > transformation that does not have that property at all: transform {1, 2} > to {1} by mapping both elements of the first to 1. It can not be > re-written as a sequence or set of transpositions. So while it is a > transformation, it does not satisfy the requirements to leave the > ordinal number unchanged.) > > So again, what is the definition of "transformation"? The definition of *allowed transformations* was given by Cantor in above text. There is *no* further question except that you are unwilling to confess your error and your insult that I was unable to understand the German text. > If it is > unrestricted (and so any re-ordering is allowed), the statement is > trivially false. It is not the question whether Cantor was wrong, but only what was the meaning of his quote. > If there are restrictions (and so not any re-ordering > is allowed), the statement might be true. But whatever, when the > first is true (any re-ordering is allowed as a transformation), the > statement is false and you can not use it, because it is not proven. > On the other hand, if there are restrictions, you have to show that > your set of transpositions form a transformation before you can use > the statement. > > > they are defined by: "welche sich zur=FCckf=FChren lassen auf eine > > endliche oder unendliche Menge von Transpositionen, d. h. von > > Vertauschungen je zweier Elemente =3D which can be traced back to a > > finite or infinite set of transpositions, i.e., interchanges of two > > elements." > > Yes, that is the definition of the property of the transformations that > leave the ordinal number unchanged. But the question remains, given > any sequence (or set) of transpositions. Is the result a "transformation"? > I do not know, because I do not know the definition of "transformation". Then read again the text in English which you quoted above. There it is stated explicitly. > > > > > Of course it is false, whether there is a sequence or not. My > > > > transpositions form a sequence. One cannot do better. > > > > > > Yes, so what? Is yuor set of transpositions a rewriting of some > > > "Umformung" in the sense Cantor implies? So there is no "of course" > > > here. We need to know what Cantor means when he writes "Umformungen". > > > > > He says it clearly: those and only those transformations which are > > realized by a finite or infinite set of transpositions, i.e., > > interchanges of two elements, do not change the Anzahl =3D ordinal > > number. > > Yes, that is a subset of the Umformungen. What is his definition of that > term? What does he mean when he writes "Umformungen"? I you are you unable to read English as well as German, try to get a copy in Dutch. > > No. Cantor obviously was correct, if infinite sets would exist. > > Again you state "obviously" when I ask for a proof. So you are apparently > not able to provide a proof but think that what Cantor writes is always > true. Who coined the term matheology? There is no proof necessary. If infinite sets exist and can be exhausted, then the set of transpositions can be finished. As no single transposition changes the ordinal number, it remains unchanged during the whole proess.. > > See above where I have written my rebuttal. You apparently have no idea > what a good definition entails. He does not define the term "Umformungen", > but defines a subset of "Umformungen" that have some particular property. > Until we know that the larger term "Umformungen" means, we have no idea > what that subset contains. The full meaning is uninteresting, as it covers even mechanical transformations. For our case only those tranformations are interesting, which Cantor defined precisiely. > > > > > > Therefore it never becomes 0 but always remains > 0 like 0.11... never > > > > becomes 1/9. > > > > > > But apparently you do not use the standard definition of 0.111... . Is it > > > a number or something else? > > > > It is a number with aleph_0 places, which are all occupied by 1's. > > Eh, it is a number that changes value? You state "never becomes". Strange > to state something like that about a number. You will have to become accustomed with that. As only potential infinity exists, not infinite process can be regarded as finished and no sequence of 1's can be regarded as static unless it is fi
From: David Hartley on 16 Jul 2006 06:45 In message <1153040519.966908.158530(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de writes > >Dik T. Winter schrieb: > >> In article <1152979000.961294.290560(a)m79g2000cwm.googlegroups.com> >>mueckenh(a)rz.fh-augsburg.de writes: >> > Dik T. Winter schrieb: >> > > In article >> > ><1152884145.033413.104250(a)75g2000cwc.googlegroups.com> muecken= >> > h(a)rz.fh-augsburg.de writes: >> ... >> Let's start with an English translation, makes it quite a bit easier for >> me (I have not to do two translations with three different languages): >> Works, p. 214, translated: >> The question, through which transformations the ordinal number of a >> well-ordered set is changed, and through which it is not changed, is >> easily answered, those, and only those transformations leave the >> ordinal number unchanged that can be rewritten as a finite or infinite >> set of transpositions, that is, of interchanges of two elements. >> Agree that this is a reasonable translation? > >OK >> >> > > No, but still, not *any* set of transpositions is permitted. Only those >> > > sets of transpositions that are the result of rewriting a transformation >> > > ("Umformungen"). So it remains interesting what Cantor is meaning when >> > > he writes "Umformungen". And when he is meaning with that word >> > >arbitrary >> > > re-orderings, he is wrong (because *all* re-orderings can be >> > >rewritten as >> > > an infinite set of transpositions). >> > >> > You do completely misunderstand what is defined and what is the >> > defintion. >> >> I think not. >> >> > Defined is Umformungen: "diejenigen und nur diejenigen Umformungen = >> > those and only those transformations" >> >> Yes, so that is not a definition of transformations, it restricts the >> transformations to a subset of them, namely only those transformations >> that have a particular property. > >Of course. "Umformungen" includes many transformations like those of >iron to hammers. > >> Apparently there are transformations >> that do *not* have that property. So the question remains, what is the >> *definition* of "transformations"? (I can easily enough give a >> transformation that does not have that property at all: transform {1, 2} >> to {1} by mapping both elements of the first to 1. It can not be >> re-written as a sequence or set of transpositions. So while it is a >> transformation, it does not satisfy the requirements to leave the >> ordinal number unchanged.) >> >> So again, what is the definition of "transformation"? > >The definition of *allowed transformations* was given by Cantor in >above text. There is *no* further question except that you are >unwilling to confess your error and your insult that I was unable to >understand the German text. > >> If it is >> unrestricted (and so any re-ordering is allowed), the statement is >> trivially false. > >It is not the question whether Cantor was wrong, but only what was the >meaning of his quote. > >> If there are restrictions (and so not any re-ordering >> is allowed), the statement might be true. But whatever, when the >> first is true (any re-ordering is allowed as a transformation), the >> statement is false and you can not use it, because it is not proven. >> On the other hand, if there are restrictions, you have to show that >> your set of transpositions form a transformation before you can use >> the statement. >> >> > they are defined by: "welche sich zur=FCckf=FChren lassen auf eine >> > endliche oder unendliche Menge von Transpositionen, d. h. von >> > Vertauschungen je zweier Elemente =3D which can be traced back to a >> > finite or infinite set of transpositions, i.e., interchanges of two >> > elements." >> >> Yes, that is the definition of the property of the transformations that >> leave the ordinal number unchanged. But the question remains, given >> any sequence (or set) of transpositions. Is the result a "transformation"? >> I do not know, because I do not know the definition of "transformation". > >Then read again the text in English which you quoted above. There it is >stated explicitly. >> >> > > > Of course it is false, whether there is a sequence or not. My >> > > > transpositions form a sequence. One cannot do better. >> > > >> > > Yes, so what? Is yuor set of transpositions a rewriting of some >> > > "Umformung" in the sense Cantor implies? So there is no "of course" >> > > here. We need to know what Cantor means when he writes "Umformungen". >> > > >> > He says it clearly: those and only those transformations which are >> > realized by a finite or infinite set of transpositions, i.e., >> > interchanges of two elements, do not change the Anzahl =3D ordinal >> > number. >> >> Yes, that is a subset of the Umformungen. What is his definition of that >> term? What does he mean when he writes "Umformungen"? > >I you are you unable to read English as well as German, try to get a >copy in Dutch. I cannot comment on the German text, but Dik's reading of the English is clearly correct, WM's wrong. To put it even more clearly Question: Which transformations preserve ordinal number? Answer: Those which are realised by a finite or infinite set of transpositions. "Transformation" is not defined here. For that matter, neither is the action of an infinite set of transpositions, nor is there any proof. If you (WM) want to use this proposition in your argument, you must fill all these gaps. > -- David Hartley
From: Dik T. Winter on 16 Jul 2006 06:55 In article <1153039371.156834.8960(a)s13g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Virgil schrieb: .... > > Every ordinal has successors, whether in ZF or NBG or most other set > > theories, and to index any of them it is sufficient to use at the > > members of any successor. > > But the successor of 4 does not consist of the ingredients of 4 only. Eh? > So the successor of all natural numbers cannot consist of only all > natural numbers only. As stated this makes no sense. But the successor of every natural number is a natural number. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 16 Jul 2006 06:53
In article <1153039038.086809.295250(a)p79g2000cwp.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1152883753.767753.301720(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: .... > > > Define 0 *+ 0 = 0, 0 *+ 1 = 1 *+ 0 = 1 *+ 1 = 1. > > > Do the following addition *+ > > > > > > 0.1 > > > 0.11 > > > 0.111 > > > ... > > > > How can I do that? You have not defined your operators on those objects. > > Yes, I know that you want to do it digit-wise on the digits. In that > > case there is still no definition for 0.1 *+ 0.11. So let us extend > > all numbers with zeros: > > 0.10... > > 0.110... > > 0.1110... > > so now we can go. But you have not defined how to do that "infinite > > sum". You use a lot of things for which you provide no defition. And, > > there is one natural number missing: 0.0... > > That is not a natural number. Depends on whether you are a follower of Bourbaki and Halmos, or of others. My learning was with Bourbaki as a basis. > The Addition > *+ was defined by "if one 1 is present the result is 1, else it is 0". > This covers also the infinite case. It does not. Define: A1 = 0.10... A2 = 0.110... etc. Now I can do: SUM{i = 1 .. n} A_i where SUM means repeated application of *+. There is nothing in your definition that covers the infinite case. > > > What is the result? 0.111... > > > > How do you get there? What rule are you using to calculate that > > "infinite sum"? But even if we allow that sum, we know that that > > is not the unary representation of a natural number. > > But it is a unary representation of a whole number, namely of the > number of digits of a real number. Nevertheless, in decimal > number of digits of a real number. Nevertheless, in decimal > representation 0.111... does exist, but not in the list. The addition > of the list numbers gives the largest number of the list. In the finite case. As you have not provided a way to do the infinite sum, I have no idea what happens in that case. > Whatever this > may be, it is not 0.111... . Proof: *Every* list number ends up with at > least one zero. Therefore there must be at least one column with only > zeros. Wrong. Provide a proof of that statement if you think it is correct. > > > This number is not among the numbers to be > > > added. But it must be, if it is different from any list number and if > > > the result of the addition is 0.111... . > > > > I see no reason for the "must be". Please first provide a proper > > definition of how that infinite operation should be performed. > > If a sum includes one or more 1's, it is 1. Therefore we have no > problem with infinitely many 1's. You have given a meaning to A1 *+ A2, A1 *+ A2 *+ A3, etc., all finite sums. You have not yet given a meaning to the infinite sum. > > Makes no sense to me. I did state: > > For all p there is an n such that An[p] = K[p]. > > You said that was wrong. I asked you why that was wrong, and you come > > up with some nonsense. I state: select n = p. Isn't it a fact that > > For all n, An[n] = K[n]? > > All n which can be indexed are in the list. K is not in the list. This is not an answer to my question. What is wrong with For all n, An[n] = K[n]? Pray, is there an n such that An[n] != K[n]? If so, what n is it? > All segments K[n] are in the list. K is not in the list. What > distinguishes K from all its segments if not at least one more 1at a > position which cannot be indexed? K has nothing else to offer. What K offers is that it has no zeroes in its representation. And all its digit positions are natural numbers. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |