Cantor Confusion
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From: Dik T. Winter on 31 Oct 2006 09:48 In article <1162299756.756951.78990(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > > 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. > > > > Sorry, but numbers are fixed and not variable. The sentence "number ... > > to come arbitrarily close" is nonsense. Numbers do not come arbitrarily > > close to each other, it is sequences that can come arbitrarily close to > > each other. > > Irrational numbers have no last digit. Therefore, with a sequence of > digits like the diagonal number is, one can never have a completed > number but only come as close as possible to any number --- or avoid to > do so. Yes. So what? The sequence comes as close as one wishes to some other sequence. What is the problem with that? > > > > 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? > > > > As many as you want. > > Not so. Of course we talk about a fixed base like 10. Ah. In that case one or two, depending on the number involved. But the diagonal obviously depends on the actual representative chosen. > > > Omega is but a convenient name of the set which exists according to the > > > axiom of infinity. > > > > 'the set' -> 'a set'. > > OK. The smallest infinite set. > > > The axiom of infinity states that there is a set with such and such > > > properties... And this set is omega. > > > > I think this is NBG. > > It is ZF too. Not implicitly. It depends on the model we are using. -- 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 31 Oct 2006 09:43 In article <1162299524.423928.41670(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > > The definition is sufficiently special (=3D non-general) to determine > > > that lim [n-->oo] {1,2,3,...,n} =3D N is correct. > > > > Yes, of course, because that was the definition you provided. So because > > it is so defined, the definition is correct. > > > > > > > > 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. > > > > "lim {n -> oo}" is an operator that works on sequences, apparently. And > > you have defined that operator for precisely one sequence of sets. > > It is defined for infinitely many sets of integers. (Infinitely many sequences of sets.) > lim [n-->oo] {-k,-k+1,..., 0, 1,2,3,...,n} = {-k,-k+1, ...,0} u N for > every k e N. > lim [n-->oo] {k, k+1, k+2,...,n} = N \ {1,2,3,...,k-1} for every k e N. Finally you give a definition. Why did it take so long? > > > > > 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. > > > > No. The second is never valid, because the limit notation is defined > > in such a way that the limit point is *never* reached (all definitions > > of limits in mathematics are formulated in such a way that the limit > > point itself, or possible function values at the limit point, are not > > used). And in mathematics 1/oo is *not* defined. > > Not in mathematics. But in a theory which assumes omega to be a whole > number. Which theory? > Die Anzahl einer unendlichen Menge [ist] eine durch das Gesetz der > Z?hlung mitbestimmte unendliche ganze Zahl. (G. Cantor, Collected > Works p. 174) > ... kann also omega sowohl als eine gerade, wie als eine ungerade Zahl > aufgefa?t werden. (G. Cantor, Collected Works p. 178) Where in the above quote is 1/oo defined? > > > > 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. > > > > Yes, so what? Set theory is more basic than analysis. In set theory > > limits are in general not defined. > > Therefore we have there limit ordinal numbers? Yes. So what? Limits are in general not defined. Especially limits of sequences of sets are in general not defined. > > > So apply this knowledge to the case of the balls and the vase. > > > > it can not be applied to it. > > What a lucky accident! > Don't you see that the whole aim of Newspeak is to narrow the range of > thought? In the end we shall make thoughtcrime literally impossible, > because there will be no words in which to express it. (George Orwell > in "1984") Nonsense. Suppose I define an ordering relation on sets. How can I apply that knowledge to numbers? Defining something in one context does not make it immediately applicable in another context. -- 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 31 Oct 2006 09:54 In article <1162300306.820436.173330(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > > 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. > > > > Again, this statement is completely in contradiction with the axiom of > > infinity. If you want to talk about mathematics without that axiom, > > please do so, but do not suggest that you are talking about mathematics > > with that axio. > > I use the simple fact that aleph_0 or omega is a number which is larger > than any natural number. No. You are using more. Primitive knowledge based on nothing at all. > A list of all finite words has no word which has omega letters. Right. > Therefore it is impossible to exchange omega letters in a diagonal. Wrong. For each element of the list a digit is calculated in the diagonal. As there are infinitely many (omega) elements in the list, there are infinitely many (omega) digits in the diagonal. > The > diaogonal cannot be roader than the list. The length of the diagonal is > the minimum of width and length. This knowledge is prior to your > axioms. Width and length are equal. > > > All finite sequences are countable. They yield another finite sequence. > > > Hence they are uncountable. Contradiction. > > > > The statement "they yield another finite sequence" is wrong in the > > context of the axiom of infinity. > > A matrix with width A and length B has a diagonal which has min(A,B) > elements. If your axiom contradicts this, then the axiom contradicts > mathematics and should be abolished. That is not contradicted. Width and length are equal. > > > 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. > > > > Forget that arguing with negation of the axiom of infinity. I am talking > > in the context of that axiom. Due to that axiom there is a set of > > natural numbers (all finite), that is itself not finite. > > And if you can conclude that in this context every straight line > crosses itself 17 times, then you will also take that as a fact? You, if that follows from some axiom, it would really be possible, unless the added axiom leads to an inconsistencey. But I think there might be surfaces where that is even valid. -- 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 31 Oct 2006 10:02 In article <1162300936.776151.45540(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > > Attention: 0.111... has only finite initial segments - and nothing > > > more. Only those can be indexed. > > > > You keep stating that, without proof. That it has only finite initial > > segments, I agree. Not the remainder. > > It is you who denies proof. Give an example of a digit which does not > belong to a finite sequence. If you cannot do so, then every digit > belongs to a finite sequence. I do *not* deny that. But there are infinitely many digits, each being part of a finite segment. > The sequence 0.111... consists of every > digit but not of more. Therefore, there is not need and no use of > talking about infinite sequences. How can those infinitely many digits form a finite sequence? > > > 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. > > > > But I never argued that. > > But it was argued in connection with my infinite binary tree. No, something quite different was argued there. Namely that the limit *also* is the number of edges in the infinite tree (or somesuch) requires transfinite induction. > > > > 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, > > > > And indeed. Induction will *not* show that 0.111... is rational. > > Induction will show everything that *can* be shown. Up to every finite > position 0.111...1 is rational. More is not possible. Irrational > numbers don't exist, but that is another topic. Yes, you again deny that the infinite exists. Again denying the axiom of infinity. While still maintaining that you are arguing within the axiom of infinity. With the axiom of infinity it exists, and irrational numbers exist, 0.111... does exist, and induction can not show that 0.111... is rational. > Anyhow, in the binary tree there is no transfinity required. There is. Namely to show that the number of edges in the infinite tree is equal to the limit of the number of edges in the finite subtrees. -- 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 31 Oct 2006 10:08
In article <1162301133.288356.234700(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1162139168.281239.183260(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > > > 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. > > > > > > It does not exist as a number. It has no b-adic representation. It > > > exists as a geometric entity. It is an idea. > > > > It exists as the continued fraction [1,2,2,2,2,...]. It also exists > > as 1 in the base sqrt(2) notation. > > And it exists in many other disguising. But it does not exist as a > number which can be put in trichotomy with all other numbers. Why not? Anyhow, I presume that you use linear algebra on occasion and eigenvalues of matrices. How can you use things that do not exist? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |