Cantor Confusion
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From: mueckenh on 31 Oct 2006 07:53 David Marcus schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > Dik T. Winter schrieb: > > > > > Euclides was apparently smarter than Cantor in this. He used the > > > parallel axiom (postulate) for something he could not prove from the > > > other axioms. > > > > But which is obviously possible and correct under special > > circumstances, while Zermelo's AC is obviously false under any > > circumstances. > > What do you mean an axiom is "false"? An axiom leading to a contradiction when added to a set of axiom which do not so, is false. The axiom that every straight line crosses itself is false in Euclidian geometry. > Are you discussing philosophy or > mathematics? I don't know what predominates in Cantor's theory - philosophy or theology, but I am sure that there is no mathematics there. Regards, WM Regards, WM
From: mueckenh on 31 Oct 2006 07:55 Lester Zick schrieb: > On 29 Oct 2006 13:35:46 -0800, mueckenh(a)rz.fh-augsburg.de wrote: > > > > >Dik T. Winter schrieb: > > > >> "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). > > > >Really? I never heard of that. Is here anybody who learned that too? It > >would interest me. No polemic intended. > > I have also heard that. In the early days of computing it was a > significant issue as to whether zero was only positive or there was > also a negative zero. These days zero is only regarded as positive and > the so called negative zero is considered an overflow situation to the > best of my knowledge. Thanks. I understand that computers have their own necessities. Regards, WM
From: mueckenh on 31 Oct 2006 07:58 Dik T. Winter schrieb: > > The definition is sufficiently special (= non-general) to determine > > that lim [n-->oo] {1,2,3,...,n} = 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. 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. > > > > 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. > > 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. Die Anzahl einer unendlichen Menge [ist] eine durch das Gesetz der Zählung mitbestimmte unendliche ganze Zahl. (G. Cantor, Collected Works p. 174) .... kann also omega sowohl als eine gerade, wie als eine ungerade Zahl aufgefaßt werden. (G. Cantor, Collected Works p. 178) > > > > 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? > > > > 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. > > 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") Regards, WM
From: mueckenh on 31 Oct 2006 08:02 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. > > > > > 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? > > As many as you want. Not so. Of course we talk about a fixed base like 10. > But if you want to make the diagonal argument to > work it makes sense to select particular representatives for each > equivalence class. So selecting for some a base pi notation and for > others a base (1 + i) notation is not likely to work. But again (and > we have discussed this before), in my opinion, Cantor's original > diagonal proof was not about the reals at all. It was about the > theorem he states in the first part of the paragraph for which you > wish only to quote the second part: "there are sets that have larger > cardinality than the naturals" and uses the set of infinite sequences > of two objects, called 'm' and 'w'. Let us drop that. How many different representatives can be chosen in Cantor's list (which needs a fixed base) for one equivalence class? > > > > 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. Regards, WM
From: mueckenh on 31 Oct 2006 08:11
Dik T. Winter schrieb: > In article <1162135573.031943.175480(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > > > In article <1161861368.657796.161130(a)k70g2000cwa.googlegroups.com> muecke= > > nh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > > > > > > > As you allow constructions by lists, and as there are uncountably ma= > > ny > > > > > 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. > > Again correct. But there are uncountably many sequences of decimal digits > that can not be defined. Where are they? Do they exist at some place in the universe? In your and my head they do not exist. > > > > > 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. > > Yes, that is what they are called. Turing even provided a precise > formulation of what a finite definition actually was. > > > > > 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. > > 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. A list of all finite words has no word which has omega letters. Therefore it is impossible to exchange omega letters in a 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. > > > > 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 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. > > > > > 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. > > 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? Regards, WM |