An uncountable countable set
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From: Virgil on 16 Jul 2006 13:32 In article <1153052131.852951.273540(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > 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? > > > > The successor of 4 is 5, and 5 = 0.11111 which cannot be indexed by 4 = > 0.1111. 5 in unay representation consists of one more 1 than 4. Hence, > 0.111... as the succesor of all naturals must consist of more 1's, than > any natural, if it is to be a number. As it is not to be a natural number, that is no problem, but it is to be, in a sense, a cardinal number, > > The successor of all naturals is not a natural and, therefore, must be > larger (because it is not less). There is no such thing as the successor of all naturals any more that there is a single successor common to both 3 a and 6. One has the set of all naturals, and that set can have a successor under the definition that the successor of any set, x, is (x union {x}).
From: Virgil on 16 Jul 2006 13:35 In article <1153052741.864854.24540(a)s13g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > > > 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. > > 0 was invented less than 2000 years ago. Natural numbers need not be > invented. However, Bourbaki and Halmos tried this trick in order to > prevent set theory from too easily been demasked as inconsistent. Don't forget John.
From: Virgil on 16 Jul 2006 13:37 In article <1153066521.156398.137690(a)s13g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Hartley schrieb: > > > >Obvious is that no transposition of two elements of a well-ordered set > > >can destroy the well-order. > > >Obvious is that, if infinite sets do exist and can be exhausted, the > > >set of my transpositions does exist and can be exhausted. > > >That's all needed. > > >Belief in anything else is matheology. > > > > Obvious is, you haven't the faintest idea of how to prove your claims. > > > Obvious is that the finite or infinite binary tree cannot contain more > paths than edges. What is obvious to "mueckenh" is here provably false to everyone else, and what is false here to "mueckenh" is provably true to everyone else.
From: David Hartley on 16 Jul 2006 14:43 In message <1153067531.577059.192510(a)m79g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de writes > >David Hartley schrieb: > >> In message <1153051856.174981.269510(a)35g2000cwc.googlegroups.com>, >> mueckenh(a)rz.fh-augsburg.de writes >> > >> >David Hartley schrieb: >> > >> >> 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. >> > >> >But "such transformations which do not change the well-order" are >> >defined here. And only those are interesting. >> >> They are not defined. They are characterised within the class of all >> transformations, but transformations themselves are not described at >> all. To use Cantor's proposition, to show that a particular >> set-theoretic "object" is an order-preserving transformation, you would >> need to show it can be written as a set of transpositions *and* that it >> is a transformation. Nowhere - in the sections you've quoted - does he >> say what a transformation is. He presumably knew what he meant; he may >> well have written a definition somewhere else, but you do not seem to be >> able to find it. > >Cantor was often redundant. Because he was very eager to get understood >quite well. But he did never defined what "is" is or what "to write" >means. Have you got me? In all of his collected works he does not >define "Umformungen" because there is nothing do define. An "Umformung" >is just a re-ordering. And such re-orderings which consist of a set of >transpositions, were *defined* by him. As I and others keep reminding you, it doesn't make any difference what Cantor actually meant. Quoting Cantor does not constitute proof. If you want to make your supposed proof of inconsistency rigorous, you must: a) define what you mean by the result of applying an infinite sequence of transpositions, b) prove that your particular sequence re-orders a well-ordering of the positive rationals to the usual ordering, c) prove that such a sequence cannot alter the order-type. (Naturally, you will do this within a standard formulation of modern set theory.) Dik and I think you can do b but not c. Virgil thinks you can't do b, but until you provide a your claims are "not even wrong", just meaningless. I'm going to stop now. It was fun contemplating the re-orderings, but it's pointless to continue arguing against your illogic. -- David Hartley
From: Franziska Neugebauer on 16 Jul 2006 14:51
mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > >> You may recall that every sequence member has trailing zeros. > > Indeed. I recall. Therefore the linearity of the list numbers enforces > a column with zeros. Nice try, but non sequitur. There is no such column. Otherwise show one or prove its existence. >> > As the first case, call it mixed number, is excluded by the >> > definition of the list, >> > there is only the >> > second case remaining, call it linear number. >> >> This is also "excluded" by the definition of the list. So it's >> pointless to discuss it either. > > Then the sum cannot be 0.111... Or "forall" must have a changed > meaning (because, you know, there are no mixed numbers with zeros > jumping around). But the *-sum _is_ as has already been formally shown 0.111.... And no, forall does not have a "changed meaning". >> > Hence (which means the >> > same as "therefore, for this reason, thus, that is why, ...") there >> > are only two logical alternatives: >> > Either 0.111... is in the list, or the *+ sum of the list numbers >> > is not 0.111... . >> >> 0,111... is not "in the list" by definition of the list _and_ >> the *-sum of the list still is 0.111... >> >> This is neither inconsistent nor contradictory. > > That is your pretension, but this claim claims a gobbledygook. What exactly do you want to posit? >> > You claim a third alternative which is prohibited by tertium non >> > datur. >> >> My "list of claims" is finite and already finished. There is no third >> alternative. > > That is what you pretend, but this pretension pretends a mumbo jumbo. What exactly do you want to posit? >> > This can be proven in another way too: >> > 1) If you remove all numbers with more than n 1's, then every line >> > of the list contains at least one zero. Mixed numbers are not >> > present, hence (which means the same as "therefore, for this >> > reason, thus, that is why, ...") the zeros mount up at the right >> > hand side. There must be a column with zeros. >> > And there are only n columns which give the *+ sum 1. >> > >> > 2) If you reintroduce a number larger than n, then the number of >> > columns with 1's grows. >> > >> > 3) If you reintroduce all numbers with a finite natural number of >> > 1's, then the number of columns grows further, but there are only a >> > finite natural number of columns with 1's. You said all these >> > numbers carry trailing zeros. But now you refrain from this >> > all-quantifiyer which according to he definition of *+ sum would >> > yield a sum zero, but assert that the 1's of these numbers cover >> > all columns. >> > >> > 4) If you at last introduce 0.111... in the list (as the last line >> > or as the diagonal number, each of its 1's in a separate line), >> > then the *+ sum remains he same. As 0.111... carries more 1's than >> > all numbers in the list (i.e., not less than and not equal to any >> > other) it is odd that its introduction should not have an effect. >> > >> > Concluding, you can assert further, that all these deductions are >> > in vain because infinity could not be refuted as little as the >> > existence of God can be refuted. And if you refrain from logical >> > deductions you are right. But if infinity were a number and would >> > behave like a number, it could be refuted. Hence your improvable >> > infinity is completely outside of mathematics, it is not a number >> > and is not good for anything else. >> >> I am not going to waste my precious time on scrutinizing this futile >> mess. > > You could utter this fully justified with respect to set theory. > Is it in particular point 4 which excites your fury? I will not read it. F. N. -- xyz |