Cantor Confusion
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From: Dik T. Winter on 28 Oct 2006 22:01 In article <gld7k21vutgpcr78bt42a6490mvqht1vb1(a)4ax.com> Lester Zick <dontbother(a)nowhere.net> writes: > On Sat, 28 Oct 2006 00:36:01 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> > wrote: > > > However one can certainly show the square root of 2 without > > > transfinity through rac construction even though its decimal expansion > > > is infinite. > > > >You need not tell that to me. You should tell that to Wolfgang > >Mueckenheim who insists that sqrt(2) does not exist because it is > >impossible to know all the decimals in its decimal expansion. > > Well you know I talked to WM about this a year or two ago and of > course I disagree with him in general terms. But an interesting aspect > of the problem popped up just a few days ago when another person > suggested that mathematical definitions have to designate a "domain of > discourse" in what appear as particular terms such as card(x) = . . . In all discussions you need a "domain of discourse". This, however, does not mean that you need to flag every term with the "domain of discourse". Otherwise you need to flag with each term whether the "domain of discourse" is the English language, the French language, or perhaps French translated to English. When I read message in this newsgroup, I generally associate "positive numbers" with "numbers larger than 0". Because I understand the main domain is Anglo-Saxon mathematics. This is in contradiction to what I did learn at university (0 is both positive and negative). But if you are reading mathematics books in English translated from French you need to be aware of this, because the main domain of discourse has change from Anglo-Saxon mathematical terms to translated French mathematical terms. (For some in this group Bourbaki is a filthy word, but the group did a lot to put together quite a bit of mathematics.) So, when I use the word positive in this newsgroup I will in general not flag it as Anglo-Saxon, because the main domain of discourse here is Anglo-Saxon mathematics and mathematical terms. The same holds for the Godbach conjecture. When you read current accounts, they are clear, but when you read the original, it is nonsense. Until you realise that the original was written when 1 was considered to be prime. According to the tables of D. H. Lehmer, there are five primes smaller than 10. You need some assumptions about the domain of discourse in which the terminology sits. In general that is clear, but some people bemuddle that and try to proof some inconsistency of (say) ZFC using something that is completely false in ZFC, and can not be proven within that theory. Also Wolfgang Mueckenheim can talk about sqrt(2) not really being a number, but if he does show he should know that he is not talking within some standard theory, but only in a theory of his own. That theory may be valuable, but it is not known whether that theory is valuable until it has been developed sufficiently. > Now without debating the issue of mathematical definition in general I > will say that if mathematikers insist on definitions drawn exclusively > in particular terms then WM may have some basis for his conclusions in > that the infinity of mathematical expressions cannot be accommodated > in a physically limited universe. He is in a sense right, of course. But that is even true when you do not draw exclusively in particular terms. The number of mathematical expressions and inferences always will be finite. What this does not mean is that all expressions and inferences are only about the finite. The whole point about using definitions in particular terms is that everybody should know what you are talking about. When you use terms without such (implicit or explicit) definitions you are likely to be misunderstood. > In other > words if mathematics insists its definitions can only be valid if cast > in particular terms of specific domains to which they apply and there > can be no infinite domains in a finite physical universe then math > definitions cannot apply to infinity. At least that's how I would > argue the issue. But mathematicians do in general not even consider the physical universe. Mathematics is not about the physical universe. It can be *applied* to the physical universe, and there are branches of mathematics that do such applications. > Now my solution to the problem would be to adopt a method of general > definition in mathematics instead of the particular method math seems > to prefer. Then if properly drawn mathematics definitions could > certainly apply to infinity provided mathematics definitions don't > need to specify any specific domain of discourse to establish the > scope and validity of its definition. For one thing, I always have stated, and maintain, that a definition in mathematics is *always* valid. It is possible that there is nothing that satisfies the definition, but the definition remains valid. The most problematical appear always to be the inference rules: for all a with property A, b holds the common (non-mathematical) interpretation being that there is at least one a with property A. The mathematical interpretation is that there is not necessarily an a with propery A (in general, there are branches of mathematics where that is false, but, again, you need a domain of discourse). But this is more in the sect of mathematics called logics. So in mathematics the statement: all partly pink elephants have green spots on their trunk is in general true. (And it is also true if you have seen "The Party".) -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: mueckenh on 29 Oct 2006 10:21 Dik T. Winter schrieb: > In article <1161860460.122685.294590(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > > > > > > > > lim [n-->oo] {-1,0,1,2,3,...,n} = N > > > > > > > > > > > > is obviuously wrong too. > > > > > > > > > > Depends on how you define the limit. > > > > > > You did not state why that one was obviously wrong. > > > > You think it must be proved that -1 is not a natural number? As it must > > be proved that an always increasing positive function does not converge > > to zero? > > No, you must tell me what that limit is, until now you have not provided > any definition of a limit of sets so I can verify whether the above is > true or false. Until now you have defined the limit of exactly *one* > sequence of sets, but that definition is not sufficiently general to > determine whether the above statement is true or false. The definition is sufficiently special (= non-general) to determine that lim [n-->oo] {1,2,3,...,n} = N is correct. > But this is again different, you state here that N is the set of natural > numbers, and you define the limit of a particular sequence of sets as > being that set N. But it does *not* cover arbitrary sequences of sets. > What, according to that definition, is: > lim {n -> oo} {-1, 0, 1, ..., n} > the definition does not make that clear. So the definition is not about > the "operator" but about a particular sequence of sets. The operator "lim [n-->oo]" defines N. In your example lim {n -> oo} {-1, 0, 1, ..., n} we have N too but in addition the numbers -1 and 0. > > > In the first > > case, we have 1/n < epsilon for every positive epsilon and we may > > *define or put* > > lim [n-->oo] 1/n = 0. > > In the second case we have without further ado > > lim [n-->oo] 1/n = 0. > > That is the difference between potential and actual infinity. > > Well, in mathematics the first form is valid, the second is not valid. If actual infinity is assumed to exist, then the second case is valid. > But now you are talking in analysis where limits are properly defined > (because there is a topology and a metric). Set theory was discovered and defined as being based on analysis. Cp. Cantor's first proof. > > > > > > That requires proof. > > > > > > > > LOL. Idle discussion. > > > > > > Perhaps, but you use it as argument, and I want to know whether it is a > > > valid argument, and so I want to see a proof. > > > > How do you know that the infinite set omega = N, which contains {n} or > > n+1 if it contains n, does have a positive number of elements? How do > > you know that it has more than one element? > > You do not know that (and can not prove it), until you have *defined* the > ordering relations. Once you *have* the ordinal numbers, you can define > the ordering relations, and a valid definition would be the following > (assuming the letters are ordinal numbers): > (1) a = b if there is an order preserving bijection between a set with > ordinal a and a set with ordinal b > (2) a <= b if there is an order preserving injection from a set with > ordinal a and a set with ordinal b > from these you can define: > (3) a != b iff not a = b > (4) a < b iff a <= b and a != b > (5) a > b iff b < a > (6) a >= b iff b <= a > Now you have an ordering. You have to show that it defines an ordering > relation indeed (not so very difficult). And as there is an injection > from {} to N, we know that {} <= N (from 2), and as there is no > bijection from {} to N, we know that {} != N (from 1 and 3), and so by > (4), 0 < omega. So apply this knowledge to the case of the balls and the vase. Regards, WM
From: mueckenh on 29 Oct 2006 10:22 Dik T. Winter schrieb: > In article <1161856793.990116.183680(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > > > Within the *real* numbers the limit does exist. And a decimal number is > > > nothing more nor less than a representative of an equivalence classes. > > > > So we are agian at this point: The real numbers do exist. For the real > > numbers we have LIM 10^(-n) = 0. Therefore, i ther limit n--> oo tere > > is no application of Cantor's argument. > > And again we are back at this point. You do not comprehend what I am > writing. The reason real numbers exist is that there are sequences of > rationals that come arbitrarily close to each other. The reason that > a decimal expansion is a representative of a real number is because the > sequence of finitely terminations of that number is a sequence of > rationals that comes arbitrarily close to other such sequences and so > falls in an equivalence class. *No* limit is involved in all of this. For the diagonal number of Cantor's list it is nnot sufficient to come arbitrarily close to a number which is different from any list number. > > > > I do not understand why you want to tie that list in with the definition > > > of the reals? > > > > > Because the real numbers are used in that list. > > Representatives of the equivalence classes that actually are the real > numbers. How many different representatives can be chosen in Cantor's list for one equivalence class? > > > > > But you should try to understand. In Cantor's list, there are > > > > limits of sequences, not equivealence classes. > > > > > > The list is a list of equivalence classes. In the decimal case > > > representatives of those equivalence classes are used. > > > > And that are limits. > > No, they are not, they are sequences of rational numbers. In Cantor's list there are omega irrational numbers. > > > > > > Where in the construction above did I use the limit omega? > > > > In using any infinite sequence already > > Oh. I did not know that I did use the limit omega there. Cantor uses it. Otherwise everything was finite. > > > > > You will need it in order to construct a real number and its decimal > > > > representation for a Cantor list. > > > > > > No. By the theory, each decimal number is a representative of an > > > equivalence class of sequences of rational numbers. By the construction > > > we get another decimal number that is also a representative of an > > > equivalence class of sequences of rational numbers. No omega is needed. > > > > Why then do you think omega is needed at all in mathematics? > > Because it comes in handy on many occasions. What *is* needed is the > axiom of infinity, because that guarantees that you can even talk about > infinite sequences. Omega is but a convenient name of the set which exists according to the axiom of infinity. > (And ultimately about limits.) Without that axiom > there is no way to prove that infinite sets do or do not exist. Correct. And with this axiom the first infinite entity which is proven is omega. And then the digits of any irrational are enumerated. Their number is omega. > > > Real > > numbers without the axiom of infinity would be a nice construction. > > I would not know how to do that. Perhaps that is possible, but in that > case you would need to reformulate the limit concept. Why then do you try to dispute that omega does play a role in constructing he reals? The axiom of infinity essentially says: There is the set omega. > > > Omega is introduced only by the axiom of infinity. > > No. It is *defined* using properties obtained through the use of that > axiom. In some fields of mathematics you do not need omega at all, but > only the axiom of infinity (analysis, and I think also algebra, number > theory, and a host of other fields). > The axiom of infinity states that there is a set with such and such properties... And this set is omega. Regards, WM
From: mueckenh on 29 Oct 2006 10:26 Dik T. Winter schrieb: > In article <1161861368.657796.161130(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > > > As you allow constructions by lists, and as there are uncountably many > > > lists, there are also uncountably many WM-constructible numbers. > > > > Where are those list? How are uncountably many different lists are > > constructed by countably many words? > > Any infinite sequence of decimal digits is a list. Correct. And this sequence has to be defined somehow. But there are only countably many definitions. > > > > Yes, that is precisely what I wrote, quite sometime ago already, about > > > the computable numbers. It is easy to show that the computable numbers > > > are countable. But the theory has been developed a bit since then. > > > The problem with what K=F6nig wrote is that not every finite definition > > > also gives a number. > > > > If you mean that there are more finite definitions than numbers, you > > are right. Therefore we have a countable set of constructable numbers. > > Pray, refrain from using non-standard terminology. WM-construcatble = > computable. Definiert man die reellen Zahlen in einem streng formalen System, in dem nur endliche Herleitungen und festgelegte Grundzeichen zugelassen werden, so lassen sich diese reellen Zahlen gewiß abzählen, weil ja die Formeln und die Herleitungen auf Grund ihrer konstruktiven Erklärungen abzählbar sind. (Kurt Schütte, Hilbert's last pupil) Finite definitions! Call this set the set of computable numbers. > > > > When you substitute "computable" for "constructable", both are indeed > > > right. But Cantor was wrong. It is only after Turing that this was > > > solved. The finite definitions are countable, but not every finite > > > definition gives a number. So there exists a list of finite definitions > > > (this has been formalised using Turing machines), but this is not a > > > list of computable numbers, because there will be non-halting Turing > > > machines in the list, and you do not know how to separate them from > > > the rest. Look up the halting problem. There is no computable list > > > of computable numbers. > > > > Why do you insist on this obvious fact? It does not support your > > position. There is a list of all finite constructions or definitions > > (encoded by numbers, Gödel). This is the definition of countablity. > > I thought that Gödel encoded theorems? Encode theorems as well as formulas for constructing numbers. Encode everything which can be expressed by a finite sequence of letters. Then build the diagonal number. This is also a finite sequence of letters, because it cannot be longer than the finite list numbers. > But it is trivial (you do not > need Gödel for that) to find that the number of finite definitions is > countable. All finite sequences are countable. They yield another finite sequence. Hence they are uncountable. Contradiction. > > > The diagonal number of this list shows the listed numbers are > > uncountable. Contradiction. > > But indeed, there exist such a list of finite definitions, but not every > finite definition yields a computable number. Some of those definitions > do not yield a number at all, so you can not even take the diagonal of > the numbers defined with such a list. Getting back to the definition > of computable number: a computable number is encoded by a Turing machine > that, when it is given a natural number n, responds with the n-th digit > of the number encoded by it. Taking the diagonal means that you can ask > the n-th Turing machine the n-th digit of the number it represents. But > how can you find that digit if the n-th Turing machine does not halt when > asked for the n-th digit? Forget the Turing machines. The diagonal of the list of all finite sequences (words) of a finite alphabet is a finite sequence because the diagonal cannot have more places than the words in the list. Regards, WM
From: mueckenh on 29 Oct 2006 11:03
Dik T. Winter schrieb: > > No. Any real number has only finite digit positions, according to Dik, > > who rigorously denies that 0.111... cannot be indexed completely by > > natural indexes. Now you see that this opinion leads to induction and > > to a contradiction. So you hurry to switch to transfinity. > > Pray. No, you are wrong. 0.1 is rational, so is 0.11, and so on, so > for every *finite* n, 0.1...1 (n-times) is rational. This does *not* > show that 0.111... is rational. All 1's can be indexed in 0.1 and in 0.11, and so on. This does *not* show that all ones in 0.111... can be indexed. > > > Indeed, i is not simple. Do we need transfinite induction for all the > > reals? Then, in fact, the natural numbers are not sufficient. Then > > some digits of 0.111... are undefined. Or do we need transfinity only > > in special cases, namely then when otherwise set theory would be > > smashed? > > You need transfinity when you want to show that something that holds in > the finite case also is valid in the infinite case. Induction will not > show that 0.111... is rational, it can only show that all the finite > initial parts are rational. In Germany children learn how to divide 1 by 9. After having gotten that, they know hat 0.111... is rational. Attention: 0.111... has only finite initial segments - and nothing more. Only those can be indexed. > And I again note that the notation 0.111... > (in the decimals) has only meaning due to the definition of that notation. You need no transfinity to show that lim [n--<oo] 1/2^n = 0 and that lim [n--<oo] (1 - 1/2^n) / (1 - 1/2) = 2. Because that was known before transfinity was introduced. You deny that omega is required to define 0.111.... But you insist that transfinite induction is required to write down the result of 1/9 in decimal representation? I did not yet know that this system is that difficult. Regards, WM |