Cantor Confusion
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From: Han de Bruijn on 6 Dec 2006 08:28 Franziska Neugebauer wrote: > Let me summarize: You are mostly off topic. Let me summarize: you are mostly not listening. When WM uses metaphors, for the purpose of convincing you of something, these would have been picked up, and interpreted properly by most normal people. But not by people with a mathematics (mis-)education, of course, as I've observed more than once in this newsgroup. Being incapable of common speech and common sense and ... being proud of it. Han de Bruijn
From: William Hughes on 6 Dec 2006 08:32 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > > And there is no natural number x, which is not element of a line. > > > > Let us write this out, taking care to note > > that line n contains all the elements of any line > > less than n. > > > > We have > > > > For every natural number n there exists a line L(n), such that > > every natural number m <= n is an element of L(n) > > > > You cannot simply exhange the quantifiers to get > > > > There exists a line L, such that for every natural number n, > > every natural number m<=n, is contained in L. > > I can simply do that for any finite linear (= totally orderd) set. You have made this comment several times but you have never said *why* this is true. You need to give some argument to show why something that is not true in the general case is true for a totally ordered set. Counterexample: Consider A= [0,1), the set of real numbers in greater than or equal to 0 and less than 1. This set is totally ordered. This set is composed of finite elements. We have the true statement For every element r in A, there exists an element s in A such that r < s. However, if we simple reverse the quantifiers we get the false statement There exists an element s in A such that for every element r in A, r<s. So the fact that you have a totally ordered set consisting of finite elements, does not mean that you can reverse the quantifiers. > So I can do it for every line. > > You need to give some other argument to show that L exists. > > This you have not done. > > Recall: The potentially infinite set of natural numbers exists. It is possible to have a bijection involving a potentially infinite set. > > It is in fact easy to show that L cannot exist. > > > > If X is such that for every natural number n, n is > > an element of X, then X is a potentially infinite set. > > > > No line L is a potentially infinite set. > > > > Therefore, there does not exist a line L such that > > > > for every natural number n, every natural number > > m <= n is an element of L. > > Therefore, there is no set of all finite lines, i.e., there is no set > containing all natural numbers as elements. However the potentially infinite set of all finite lines exists. It is possible to have a bijection involving the potentially infinite set of all finite lines. > > > > > > The fact that there each natural > > number is a member *some* line, does not > > mean there is one line which contains all > > natural numbers. > > > > The point remains. > > > > A set with a largest element can have elements > > all of which are natural numbers. > > > > A potentially infinite set without a largest element > > can have elements all of which are > > natural numbers. > > Correct, although this set can never be considered complete. Irrelevent. It does not matter whether we can consider a potentially infinite set "complete" or not. All that matters is that we can have a bijection involving a potentially infinite set. > > > > There cannot be a bijection between a set with a largest > > element and a potentially infinite set without a largest > > element. > > > > Therefore there cannot be a bijection between the > > diagonal and a line. > > > > Do you intend to keep claiming that a bijection can exist > > between the diagonal and a line?. > > I claim that every element of he diagonal must be an element of a line. The statement Every element of he diagonal must be an element of a line. is true. But the statement Every element of he diagonal must be an element of a single line. is false. Therefore we cannot use the statement Every element of he diagonal must be an element of a line. to conclude that there is a bijection between the diagonal and a single line. Do you intend to keep claiming that a bijection can exist between the diagonal and a single line?. > > If the "infinite set of finite numbers" existed, then we would have a > bijection between the diagonal and a line. > We have agreed to disagree on whether the "infinite set of finite numbers" exists. However, we have agreed that the "potentially infinite set of finite numbers" exists. > (A diagoal cannot exist without being in a line.) False. Every element of the diagonal must be in some line. However, there is no line that contains all elements of the diagonal. So the diagonal is not in a line. The elements of the diagonal form the potentially infinite set of natural numbers. This potentially infinite set exists, so the diagonal exists. So the diagonal can exist without being in a line. - William Hughes
From: Eckard Blumschein on 6 Dec 2006 09:01 On 12/5/2006 10:29 PM, Virgil wrote: > In article <457578DE.7030505(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 12/5/2006 1:14 AM, Virgil wrote: >> > In article <457467D5.7020201(a)et.uni-magdeburg.de>, >> > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >> > >> >> On 12/1/2006 8:55 PM, Virgil wrote: >> >> > In article <45700723.3060406(a)et.uni-magdeburg.de>, >> >> > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >> >> > >> >> >> On 11/30/2006 1:39 PM, Bob Kolker wrote: >> >> >> >> >> > Division by zero in a field yeilds a contradiction. >> >> >> >> >> >> Just this contradiction resides already in the notion of (actual) >> >> >> infinity. >> >> > >> >> >> >> > Division by zero in standard sets of numbers is not defined because >> >> > there is never a unique x in such sets of numbers for which a = 0*x. >> >> > Either no x works or more than one works. >> >> > >> >> > Infinity has nothing to do with it. >> >> > >> >> > A finite example: >> >> > >> >> > The residues of the integers modulus a prime is always a finite field >> >> > under the usual addition and multiplication, so there is no >> >> > "infinity" involved, but division by zero in those fields is still >> >> > barred for the reason above, a = 0*x can never have a unique solution. >> >> >> >> I do not feel limited in thinking to the indefinitely large. I likewise >> >> consider the indefinitely small (infinitesimal). >> > >> > In finite rings, both are irrelevant, but the issue of division by zero >> > is the same even in such rings. Those who try to drag in the infinite or >> > infinitesimal in discussing the division by zero issue, just do not >> > understand the issue. >> >> Hopefully you can substantiate this pure suspicion. >> Being an engineer, I vaguely recall that a Zahlring is something like a >> loop. Let me fantasize: {i, i^2, i^3} Is this a ring? >> So far I do indeed not understand why the issue of rings matters in case >> of division by zero. > > > > Consider a set of 3 elements, say A = {x0,x1,x2} and binary operations + > and * , mappings from AxA to A that are commutative, associative, and so > that * distributes over + and such that > (1) for all a in A, x0 + a = a ,and x0 * a = x0 (x0 is a "zero" element) > (2) for all a in A, x1 * y = y (x1 is a unit element) > (3) other "additions" where a + b = b + a for all a,b in A > x1 + x1 = x2, x1+ x2 = x0, x2 + x2 = x1 > (4) other "multiplications" where a*b = b * a fora all a,b in A. > x2 * x2 = x1 > > Alternately use the addition and multiplication tables below: > > + | x0 x1 x2 * | x0 x1 x2 > ---|--------- ---|---------- > x0 | x0 x1 x2 x0 | x0 x0 x0 > x1 | x1 x2 x0 x1 | x0 x1 x2 > x2 | x2 x0 x1 x2 | x0 x2 x1 > > One may verify that { A, +, * } satisfies all of the properties of a > field. > > One may define in it a subtraction "a - b" for a and b in A > by a - b = c if and only if Card({c:a = b + c, c in A } ) = 1, > i.e., a = b + c has one and only one solution > > One may similarly define in it a division "a / b" for a and b in A > by a / b = c if and only if Card({c:a = b * c, c in A } ) = 1. > i.e., a = b * c has one and only one solution > > According to this definition, division by x1 and x2 will always be > possible, but division by the zero element, x0, will not ever be > possible. > > And the issue of "infiniteness" of a quotient is totally irrelevant. >> >> >> >> Isn't it better to understand why it is incorrect than simply to learn >> >> >> it is forbidden? >> >> >> >> >> >> Eckard Blumschein >> >> > >> >> > It is better to understand the real reason (see above), but Eckard >> >> > doesn't seem to understand the real reason. It has nothing to do with >> >> > "infinity". >> >> >> >> Not directly with the indefinitely large, yes. >> > >> > Where does the "infinitely large" or "infinitesmially small" enter into >> > finite rings, such as the fields of integers modulo a prime? >> >> I do not grasp your point. Mathematical closed loops (meshs) are of >> course pathways of infinite length. Correspondingly stars (nodes) add >> all branches to the indefinitely small (zero). > > See the finite field example above. Division is not possible in it. But > also none of this infinite stuff is relevant in it either. Electrical engineers like me benefit a lot from dualities and inversion. Let me briefly explain: Maxwell's equations are symmetric with respect to B and E. One may mutually exchange these variables. Impedance corresponds to admittance, etc. Inversion means to find reciprocal expressions, structures, and/or properties, e.g. mesh-->node, node-->mesh, oo-->0, 0-->oo. So we used to frequently equate 0 with anything/oo and oo with anything/0. I understand: While 0 is a recognised number because it can easily be reached via subtraction of something finite, oo is not likewise approachable. One would require addition of something infinte. Nonetheless, I guess, the infinitely large and the infinitely small are closely related to each other. >> >> > >> > The division by zero question has the same answer, and for the same >> > reasons, in these rings as in infinite rings. >> >> Why not? > > See above.
From: Franziska Neugebauer on 6 Dec 2006 09:03 Han de Bruijn wrote: > Franziska Neugebauer wrote: >> You still have not yet understood the concept of inifinite sets. > There may be no person in the world who understands them better than > Wolfgang Mueckenheim. ,----[ http://en.wikipedia.org/wiki/Understand ] | Understanding is a psychological process related to an abstract or | physical object, such as, person, situation and message whereby one is | able to think about it and use concepts to deal adequately with that | object. `---- He has yet only documented his ability to deal _in_adequately with that object (infinite sets). This does not support your claim that he does understand the concept. > But I think _you_ still have not yet understood the difference between > "understanding" and "being reluctant to accept". I think that he does not only not understand the concept of infinite sets but also he refuses to accept this fact. F. N. -- xyz
From: Eckard Blumschein on 6 Dec 2006 09:19
On 12/5/2006 10:36 PM, Virgil wrote: > I do not have a more apt word than fictitious in the sense >> it was used by Leibniz in order to stress the conceptual difference >> between addressable discrete numbers and merely attributed without such >> address positions. If you deny this conceptual difference, then you are >> denying the difference between generally countable rationals on one >> sinde and generally uncountable just fictitious reals on the other side. > > There is already a word, "irrational" that conveys everything that is > needed. The border line of countability separates reals from rationals. >> >> In other words: Genuine numbers are countable, fictitious numbers are >> uncountable. The latter do not have an available numerical address. >> >> >> > They are all equally fictitious, creations of the mind having no >> > existence outside of the mind. >> >> This is an attempt to hide that Dedekind and Cantor built an Utopia. > > The labels "genuine" and "fictitious" are equally an attempt to hide the > reality that your supposedly "genuine" numbers are no more genuine than > any others, They are categorically quite different. According to Peirce, fictitious numbers are mere potentialities. According to Brouwer and Pratt they do not obey trichotomy. Who denies their quite different properties will never understand what makes the rationals different from the reals. The thread nightmare real numbers has been lasting since September the 9th until now. I admit, mathematicians are not trained to grasp the difference between quantity and quality. > nor your supposedly "fictitious" numbers are no more > fictitious, than any others, in the everyday meaning of those labels. They are not approachable with a finite number of steps. They are no more number-like as is infinity. |