An uncountable countable set
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From: Franziska Neugebauer on 17 Jul 2006 18:00 Franziska Neugebauer wrote: > mueckenh(a)rz.fh-augsburg.de wrote: >> Franziska Neugebauer schrieb: [...] >> However, there is no natural numbers with this property, > > Could you precicely _define_ which /property/ you are talking about? BTW: The /property/ _of_ _which_ object? F. N. -- xyz
From: Dik T. Winter on 17 Jul 2006 18:44 In article <1153145345.281803.129520(a)35g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Franziska Neugebauer schrieb: > > mueckenh(a)rz.fh-augsburg.de wrote: .... > > > 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. > > The proof requires logic. Therefore I am afraid you will not accept it. > It reads: Either there is a column with only zeros, or there is at > least one 1 in each column spanned by the digit positions of 0.111... , > isn't it? Yes, right. And now the remainder of the proof, please? -- 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 17 Jul 2006 19:35 In article <1153147907.669663.129980(a)75g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > > > You do not believe in limits? Otherwise, why does the definition not make > > > > sense? Quote the definition for precisely that notation: > > > > 0.999... = lim{n -> oo} sum{k = 1..n} 9.10^(-k) > > > > what part of that definition makes no sense? > > > > > > "n --> oo" because there is no natural number oo. n cannot become oo > > > whether there are infinitely many natural numbers or not. Each is > > > finite. Thus, for n e |N, the result is always different from 1.000... > > > . > > > > n -> oo does *not* mean that n will become some oo, because n will not > > become oo; it only means that n grows without bound. And as you should > > know how the limit given above is defined, you ought to know that. > > For each (real) epsilon > 0 there is an n0 such that for n > n0 > > |1 - sum{k = 1..n} 9.10^(-k)| < epsilon > > at what place does n become oo? In the above, take n0 = log_10(-epsilon) > > when epsilon < 1.0, else take n0 = 0. > > n does not become oo. And therefore we have only the epsilon > definition. And therefore we have *always* undefined digits in any > irrational number and in the diagonal of Cantor's list. Sorry, we are now talking about 0.999..., please remain with the argument. You said that the definition is silly. Why is the definition above silly? And that there are undefined digits in irrational numbers does not bother me in the least. I have no idea why you have a problem with that. But to go on: > And therefore we have *always* undefined digits in any > irrational number and in the diagonal of Cantor's list. There are > *always* most of its digits unknown, i.e., always there are more digits > unknown than are known. No matter how small an epsilon you select. For Cantor's diagonal no epsilon is needed, neither is their for irrational numbers. The only thing we need to know is that the digits are all in the range [0, 9], and that is sufficient (by minoration and majoration) that the sequence converges and has (if we want a complete field) a real number as limit. And for all the four different definitions of the real numbers I know it can be shown that the definitions are equivalent (i.e. the resulting fields are isomorphic) and that the resulting field is complete. > Hence you cannot prove that the digits of the diagonal are all > different from those of the line numbers. Why not? For the proof no exhaustion is needed. Just like (in another thread) the proposition For all p there is an n such that An[p] = K[p] does not need exhaustion. In the case of the list and the diagonal the similar statement is For all p there is an n such that An[p] != D[p] proving that D is different from all An. BTW, in the first case also the proposition For all p there is an n such that An[p] != K[p] proving that K is also different from all An. In the first and second case take n = p, in the third case take n = p + 1. This is *not* a proof by exhaustion. It is simply stating a fact, with an easy proof. > You can prove it for each one > but you cannot prove it for all. If something is true for each one, it is true for all. Let's reason with the excluded middle (because that would complicate matters). Assume it is not true for all, then there must be some for which it is not true (say z). On the other hand, it is true for each one, so also true for z. A contradiction, hence the assumption is false. And in the cases above you can proof it for each one in one sweep, because you prove it for an arbitrary element. > That is the same problem as with the > *+ sum of my list. You can prove the sum is 1 for each column but you > cannot prove it for all columns. But you can. > Because stepwise exhaustion of > infinite sets is impossible. Otherwise my (and Cantor's) reordering > could be completed. If stepwise exhaustion is impossible, your reordering could be completed. But you can with your *+ operation give the proof in one sweep, but you failed to answer to my question: "what happens at infinity?". But, that was precisely what I intended when I asked you what you meant with "*+ ..." in 0.1 *+ 0.11 *+ 0.111, because you gave no definition. As you gave no meaning to that infinite "sum", only to finitely many of them (and in that case going to infinity is impossible), there was no way with your definitions. > > > But the assertion > > > that the digits of these limits could be used to construct a diagonal > > > number is simply nonsense. > > > > Only assertion and meaning. No content. > > Proven by epsilon. Set theory lives by contradiction. Some exhaustions > of nfinite sets are accepted other are not. You can exhaust an infinite set if you take out all elements at once. That is why the function f(n) = n + 1 is a bijective mapping from {1, 2, ...} to {2, 3, ...}, every definition works at once, you need not to count to 100 to know what f(100) is. And the same is the case with Cantor's diagonal number. You need not look at the first 100 elements to find the 101-st element to know that the 101-st digit of the diagonal is. That is what the definition of a list is. A mapping from N to the members of the list. Your set of transpositions are different. They do not work at once, you need sequencing (transpositions are in general non-commutative). In the case of a sequence you need limiting procedures to show what would be the "final" result (like lim{n -> oo} 1/n = 0). But the problem with limiting procedures is that the "final" result has not necessarily the same properties as each and every finite result. Let me give an example with transpositions. Let's say that (a, b) means that we interchange the a-th and b-th element in an ordering. Let us start with the ordered set of natural numbers {1, 2, 3, ...} (yes, not Bourbaki this time). Let's define a sequence of transpositions: (1, 2)(2, 3)(3, 4)(4, 5)... with a proper measure and a proper definition of limit, I think that we can show that that leads to the ordered set {2, 3, ..., 1}. Now interch
From: Dik T. Winter on 17 Jul 2006 19:41 In article <1153148551.942037.97110(a)s13g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > In article <1153052131.852951.273540(a)b28g2000cwb.googlegroups.com> muecke= > nh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: .... > > (1): 0.111... is not *the* successor of anything, we may sloppily say that > > it is *a* successor of all naturals, just like 10 is *a* successor of > > 2 > > It was Cantor who coined this expression saying " da? omega die erste > ganze Zahl sein soll, welche auf alle Zahlen nu folgt." (Works, p. > 195) So it must be larger anyhow. Yes. Not *the* successor, but *a* successor. > > > The successor of all naturals is not a natural and, therefore, must be > > > larger (because it is not less). > > > > Yes, it is larger than all naturals, but I would not call it *the* > > successor, but *a* successor, or, if you wish, *the smallest* successor. > > However, not all of its 1's in unary representation can be indexed by > natural numbers because they are smaller. Are you again arguing that the statement For all p, there is an n such that An[p] = K[p] is false, or are you arguing that the statement For all p, there is an n such that An[p] != K[p] is false? > There is no actually infinite set of finite numbers. Axiom of infinity. -- 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 17 Jul 2006 20:44
In article <1153168957.805313.57460(a)p79g2000cwp.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > > In German it is "Umformungen", that means "change of the order". Cantor > > > says: those changes, which and only which can be traced back to > > > finitely many or infinitely many transpositions of elements, each > > > transposition including two elements. > > > > So you claim that Cantor meant unrestricted re-ordering. If he meant that, > > his statement is false. That is entirely possible, of course. > > Of course. But after reading the context today, I come to the conclusion that he does *not* mean unrestricted re-ordering. The context is what ordinal numbers can a set of particular cardinality have when it is well-ordered. His transformations are (in my opinion) only transformations that change order (and not arbitrary transformations), and also that the question ("which transformations") refers to transformations that change a well-ordered set to a well-ordered set. > His other statements on infinity are wrong too. In > particular that which says that 0.111... is different from every > natural number but no digit is different from every digit of every > natural number. Strange enough, when I read what he wrote, he does not state that, I think quite contrary. Pray read page 213. Especially the sentence: "w?hrend die Mengen erster M?chtigkeit nur durch (mit Hilfe von) Zahlen der zweiten Zahlenklasse abgez?hlt werden k?nnen..." translated: "while sets of the first cardinality only can be counted through (with the help of) numbers from the second class of numbers..." I think (but have not looked it up for a thorough study) that the first cardinality refers to aleph-0, and that (of this I am sure) second class numbers are numbers of the form w, w+1, etc.. So we can infer that he claims here that to count the numbers of the set of natural numbers (cardinality aleph-0) you need at least w. I think not very dissimilar from what you are arguing. And I think that you are agreeing with Cantor. It may be noted that since that time quite a bit of research has been done to give better foundation. This has resulted in redefinitions of terms by Cantor and sometimes different (contradicting) formulations. It was a field of research in progress, and even published results could be shown to be (partially) wrong using later insight. Another thing that did strike me: on that page he assumed the axiom of choice (that is, every set can be well-ordered). > > Now you use an entirely new term: "can be exhausted". I think you mean > > that you can take out elements one by one and doing this at some stage > > the infinite set becomes empty. Howver, I think, that if that can be > > done, that there is a last element you can take out. And, according > > to the axiom of infinity, that is not possible, so infinite sets can > > not be exhausted in this sense. > > But in another sense? All elements at once? > > You use a contradiction of the axiom of infinity (can be exhausted), so > > there is no disproof. Now try the same with the assumptions that > > infinite sets do exist and can not be exhausted. > > Then there is no evidence that they exist. Again philosophical and hence quite a bit of matheology. What means "exist"? I have no idea. As a mathematician I am given a set of axioms and show the results from the set of axioms. Some have their uses in practical sense, some have no use in a practical sense at all. Give me a different set of axioms, and I will do the same. That is what happens all the time. Start with the Eclidean postulates (axioms) and show that the sum of the angles of a triangle is 180. That is a result that may or may not have its uses. Now change the parallel postulate and find that the sum of the angles of a triangle is in general not 180. But also in this case the results may or may not have practical implications. > But his position was also, that infinite sets can be > exhausted if for each element there is a precise definition like that > leading to the well-order of the rationals. The well order of the > ordinals is another example. And the transformations consisting of > transpositions is a third. You should better read his positions thoroughly, compare it to current versions of set theory, note the differences, and, after that, hit at the wrong-doer. But this is of course not mathematics. > > Well, I think *any* re-ordering can be accomplished by a sequence of > > transpositions of two elements. > > Cantor shared your opinion, but only in case of finite sets. Perhaps, I have seen no evidence of that. > > But when I write: > > The question, though which froobles the ordinal number of a > > well-ordered set is changed, and through with it is not changed, is > > easily answered, those, and only those froobles leave the ordinal > > number unchanged that can be rewritten as a finite or infinite > > set of transpositions, that is, of interchanges of two elements. > > you do not need the definition of froobles? I can tell you that my > > statement is entirely correct. (I can state that your sequence of > > transpositions does not form a frooble.) > > If you exchange "can be rewritten as" by "can be traced back to" or by > "consist of", as is the meaning of Cantor's German statement, then > "those froobles" is obviously merely another name for " fin. or inf. > set of transpositions". You are wrong. Let's suppose that a "frooble" is a transformation that interchanges the first two elements of an ordered set (I do not say it is, but let us just assume for the argument). > > Eh? Infinite sets do exist, but they can not be exhausted. So why do they > > not have cardinality? It is entirely possible to define cardinality for > > them. What is your problem with that? > > That it is not possible to do so without well-ordering all elements. > That is nothing but an exhaustion. Cardinality does not need well-ordering. It is ordinality that needs well-ordering. And, well-ordering is not an exhaustion. > > > One should have seen that earlier, then Bourbaki would not have > > > succedeed to define 0 as natural number, even in political decisions. > > > > Pray explain |