Cantor Confusion
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From: Virgil on 5 Dec 2006 15:48 In article <1165324945.385144.34720(a)79g2000cws.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > There is no reason to give up continuity for applied mathematics. Then why does WM argue that we should give it up in pure mathematics?
From: Bob Kolker on 5 Dec 2006 15:57 MoeBlee wrote: > In the sense you're trying to get across to the other poster, I > understand your point. But, just for the record, in a technical sense > in set theory, as integers, rational numbers, and real numbers are > themselves sets, it does make sense to say whether one of them is > countable or not. For example, where integers are defined as > equivalence classes of natural numbers, each integer is itself a > denumerable set. I am not necessarily endorsing anything the other > poster has said; I'm just adding the technical note that in a strict > set theoretic sense, even numbers are sets and thus it is meaningful to > talk about the cardinality of a number. an element of a ring or a semi-group is a set? Bob Kolker
From: MoeBlee on 5 Dec 2006 16:14 Bob Kolker wrote: > MoeBlee wrote: > > In the sense you're trying to get across to the other poster, I > > understand your point. But, just for the record, in a technical sense > > in set theory, as integers, rational numbers, and real numbers are > > themselves sets, it does make sense to say whether one of them is > > countable or not. For example, where integers are defined as > > equivalence classes of natural numbers, each integer is itself a > > denumerable set. I am not necessarily endorsing anything the other > > poster has said; I'm just adding the technical note that in a strict > > set theoretic sense, even numbers are sets and thus it is meaningful to > > talk about the cardinality of a number. > > an element of a ring or a semi-group is a set? If it is an object of Z set theory without urelements. Of course, I recognize that one may study algebra without taking the objects to be set theoretic entitites. On the other hand, one may study algebra from within Z set theory (so the variables range over sets, even if not specified in the case of, say, rings in general exactly which sets), as another instance of making good indeed on the notion that set theory provides for the expression of virtually all of ordinary mathematics. MoeBlee
From: Virgil on 5 Dec 2006 16:29 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. > > > > > 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: Virgil on 5 Dec 2006 16:36
In article <45757FEB.5030804(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/5/2006 12:39 AM, Virgil wrote: > > >> If irrational numbers are thought > >> to complete the rationals which sounds quite logical, then the > >> constituted entity of the reals has to be as fictitious as the > >> irrationals. > > > > And the rationals and the naturals and all other mathematical > > constructions. > > No. No. No. Yes! Yes! Yes! 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. "Fictitious" applies equally well to every number. > > 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, nor your supposedly "fictitious" numbers are no more fictitious, than any others, in the everyday meaning of those labels. |