Cantor Confusion
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From: Virgil on 6 Dec 2006 18:04 In article <4576BC30.2000101(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/5/2006 9:23 PM, Virgil wrote: > > >> Do not confuse Cantor's virtue of belief in god given sets with my power > >> of abstraction. > > > > Cantor's religious beliefs are as irrelevant as EB's beliefs in his own > > infallibility. > > I am not infallible. Show me my errors, and I will express my gratitude. > > > > >> Cantor said: Je le vois, mais je ne le crois pas. Obviously he didn't > >> infinity. > > > > What is "he didn't infinity" supposed to mean? > > Thank you for this hint. I intended to write "he didn't understand > infinity" but due to work from 6.30 to 19.30 without break I was tired. It is evident that EB understands "infinity" even less that Cantor. But as Cantor was a pioneer, he could be expected to find the frontiers of mathematics a bit confusing. EB is confused without such justification.
From: Virgil on 6 Dec 2006 18:15 In article <4576CD31.2080808(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > 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. it shows in your personality, dual and inverted. In mathematics, subtractions are all defined in terms of unique solutions to addition problems. Where no such unique solutions exist, subtraction also does not exist. In mathematics, divisions are all defined in terms of unique solutions to multiplication problems. Where no such unique solutions exist, subtraction also does not exist. This is true in arbitrary rings, even those with no notions of size of members. Thus smallness and largeness are irrelevant to whether a division can be performed. The division a/x can be performed if and only if there is a unique b such that a = b*x. In a ring with more than 1 element, that cannot ever happen.
From: Virgil on 6 Dec 2006 18:22 In article <4576D181.4040203(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > 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. It is not a line. > They are not approachable with a finite number of steps. They are no > more number-like as is infinity. One can add, subtract, multiply and divide reals as successfully as one can do it with rationals. With the Cauchy sequence model it is easy. With the Dedekind model, it is a bit more difficult, but still unambiguous.
From: Dik T. Winter on 6 Dec 2006 19:46 In article <1165421463.339178.48680(a)j44g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > William Hughes schrieb: .... > > 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 > > This set is not finite. The set in the quantifiers you are using is not finite either. The quantifier are not over a single line, but over the set of natural numbers. As William wrote: For every natural number n there exists a line L(n), such that every natural number m <= n is an element of L(n). See the infinite set in the first quantifier? -- 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 6 Dec 2006 19:54
In article <4576FD8D.60408(a)et.uni-magdeburg.de> Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> writes: > On 12/6/2006 1:14 AM, Dik T. Winter wrote: > > In article <4575B6E1.4010505(a)et.uni-magdeburg.de> Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> writes: .... > > > When I presented ideas in connection with > > > http://iesk.et.uni-magdeburg.de/~blumsche/M283.html > > > I faced scepticism or refusal as well as the hint to compactification. > > > > Sorry, I can't read Word documents (I have no reader available). > > Could you read a pdf version? Yes. > > But again who requires one-point compactification with what goal? > > > > > I consider my ideas still flawless. I even found plausible answers to > > > several questions no mathematician was able to provide a convincing > > > answer to. So I doubt about fundamentals which require compactification. > > > > I understand that no mathematician was able to provide an answer that > > could convinve you. > > It started with the question how to deal with the nil in case of > splitting IR into IR+ and IR-. I got as many different and definitve > answers as there are possibilities. I expected that there is only one > correct answer, and I found a reasoning that compellingly yields just > one answer in case of rationals and a different one in case of reals. Oh, well. In Bourbaki's mathematics R+ and R- both contain 0. So you are a follower of Bourbaki after all? But of course the 0's are the same. If they were different you would have quite a few problems with limits and continuity. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |