Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: Virgil on 4 Dec 2006 18:34 In article <45745F0A.2000408(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/1/2006 9:42 PM, Virgil wrote: > > In article <45706268.1020005(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> On 12/1/2006 1:37 AM, Dik T. Winter wrote: > >> > In article <456EEA86.20001(a)et.uni-magdeburg.de> Eckard Blumschein > > > >> > Oh, for once, try to talk mathematics. By the axiom of infinity the > >> > set of all naturals is neither hypothetical nor fictitious. > >> > >> This axiom combines flawless Archimedean reasoning with an at least > >> questionable replacement of the notion number by the notion set. > > > > When EB presents a completed axiom system from which he can generate > > mathematics, or at least arithmetic, he may join the lists, but until > > then he is merely a spectator at mathematics, and not competent to be a > > judge. > > At least, a spectator is not blind. A spectator who chooses not to see what is there to be seen, as WM keeps doing, is no better off that if he were blind.
From: Virgil on 4 Dec 2006 18:39 In article <457463BF.7010001(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/1/2006 9:35 PM, Virgil wrote: > > In article <1164982199.959381.134510(a)j72g2000cwa.googlegroups.com>, > > We do not yet /know/ that the physical universe is not continuous, so > > why should we reject a mathematically continuous real number system? > > I do not just agree. I would even like to stress once again that the > mathematical concepts of continuum is independent from its application. > What about continuous real numbers, I only vote for a little bit more > honesty and more consequent reasoning. What WM votes for, he then rejects when it is offered him. > 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. They are all equally fictitious, creations of the mind having no existence outside of the mind. Though the ideas they allow are often useful outside of the mind.
From: Virgil on 4 Dec 2006 19:14 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. > > > >> 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? The division by zero question has the same answer, and for the same reasons, in these rings as in infinite rings.
From: Virgil on 4 Dec 2006 19:17 In article <45746ACD.1020008(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/1/2006 8:27 PM, Virgil wrote: > > In article <45700481.7010300(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> Why should mathematics be esoteric? > > > > Not all of it is. Various bits of it come at various levels of > > abstraction, and even children understand the least esoteric bits. > > For my feeling, Dedekind and Cantor were lacking power of abstraction. From present evidence, they had a great deal more "power of abstraction" than EB has.
From: Virgil on 4 Dec 2006 19:20
In article <45746B98.5040606(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/1/2006 8:20 PM, Virgil wrote: > > In article <1164967792.130794.251330(a)j72g2000cwa.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > >> May be if you apply your personal definition of potentially infinity, > >> but not if you apply the generally accepted definition. > > > > What "generally accepted" meaning is that? Most mathematicians do not > > accept that a set can be "potentially" infinite without being actually > > so. > > I see it quite differently: Potentially and actually infinite points of > view mutually exclude each other as do countable and uncountable, > rational and irrational. Except that countable and uncountable coexist within the same set theory and rational and irrational coexist within the same real umber field. But potentially infinite does not exist within ZF or NBG or NF or any other standard set theory. |