Cantor Confusion
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From: Eckard Blumschein on 8 Dec 2006 10:23 On 12/7/2006 8:51 PM, Virgil wrote: > In article <4577F17F.7000005(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >> Dedekind himself let the question open whether his cut belongs to the >> left or the right side. He was also lacking insight. > > Actually, Dedekind saw that it made no difference which way it went, as > long as one one was consistent about it. Those, like EB, who fail to see > what Dedekind saw are the ones lacking insight. One one? Perhaps a typo. Dedekind's lack of insight is obvious: He imagined exemplary rationals like distinct points on the continuous line. On this basis he claimed to be able to separate all points and all reals, respectively, into left and right ones. While he may claim having defined a cut somewhere, he cannot properly and directly attribute a belonging address.
From: Eckard Blumschein on 8 Dec 2006 10:41 On 12/7/2006 8:47 PM, Virgil wrote: > In article <4577F0AD.7070802(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >> Moreover, rational numbers loose >> their property of being countable if they are embedded into the >> continuum. > > Then according to EB, finite subsets of the reals are uncountable. > Eb's misuse of mathematical terms is.unaccountable Well, I have to stress that I am referring to reals as they were homogenously considered in DA2 (all fictitious in that they were assumed to have actually infitely much of decimals). The presently mandatory reals are an inconsistent mix of irrationals and rationals. As soon as you are speaking of finite subsets, you are leaving the continuum of the reals. There ar no subsets within any piece of continuum. >> At least there is no possiblity to >> decide inside the genuine continuum whether a fictitious "element" > > How can a "genuine" continuum be made up entirely of "fictitious" > elements? What looks fictitious from one point of view looks genuine from the opposite one and vice versa. > >> The primary continuum is strictly speaking amorph. >> There is no structure available inside this continuum. > > Then it is not a mathematical object at all, Geometry and nonlinear functions are mathematics too. > as every mathematical continuum has a good deal of internal structure. You are confusing the original continuum according to Peirce with Hausdorff's pseudo-continuum. > Such non-mathematical > notions are of no interest within mathematics. Let us not hear about > them further! > > EB's anti-mathematical rants create more smoke than light. Anti-illusive is not anti-mathematical, on the contrary.
From: Han de Bruijn on 8 Dec 2006 10:42 stephen(a)nomail.com wrote: > Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > >>stephen(a)nomail.com wrote: > >>>But everything can be modelled as a set. > >>Define "everything" and prove that claim. > > By "everything", I meant everything mathematical. Of course that is not 100% precise. > And no, I cannot prove it. But so far all the various objects of mathematics can be > modelled using set theory. That is what is meant by set theory being a foundation > for mathematics. If someone were to invent something "mathematical" (whatever that may > mean exactly) that could not be described in terms of set theory, then set theory would > no longer serve as a foundation. But given that the basics such as the real numbers, > functions, limits, calculus, etc. all can be founded in set theory, it would have to > be something strange indeed. Not that there is anything wrong with strange, but you > probably would like it less than set theory. Correction. By "everything" you probably mean "everything according to nowadays mainstream mathematics", which _is_, of course, "mathematics", according to your probably rather limited view. But since you can not really prove anything of the kind, I will rest my case. Han de Bruijn
From: Eckard Blumschein on 8 Dec 2006 11:01 On 12/7/2006 7:43 PM, Mark Nudelman wrote: > On 12/7/2006 3:15 AM, Eckard Blumschein wrote: >> Just try and refute: >> >> Reals according to DA2 are fictitious >> Reals according to DA2 are fictitious >> With fictitious I meant: They must not have a directly approachable >> numerical address. This was the basis for the 2nd DA by Cantor afer an >> idea by Emil du Bois-Raymond. >> >> Good luck >> > > Can you define what you mean by a "directly approachable numerical address"? > > Do you mean that SOME reals don't have such an address, or that NONE of > them have such an address? Is "sqrt(2)" a "directly approachable > numerical address"? > > If you're saying that some reals do not have a finite decimal > representation, that's true, but so what? Why does that make them any > more "fictitious" than the integers? > > --Mark There are no single reals but just infinitely much of such fictitious elements within each interval. If it was allowed to quantify them, then one could say they are simply too much as to be countable or as to be apt for any numerical operation. Wittgenstein wrote: (I just give the gist) Too many laws correspond to lawlessness. Kronecker argued: There are no irrational numbers. Nobody did take this serious for several practical reasons. Nonetheless he was correct in principle. I would like to follow Leibniz who suggested to calculate with fictitious numbers almost as if (vf.Vaihinger) they were genuine ones.
From: Eckard Blumschein on 8 Dec 2006 11:05
On 12/7/2006 7:07 PM, Lester Zick wrote: > On Thu, 7 Dec 2006 02:18:36 -0500, David Marcus > <DavidMarcus(a)alumdotmit.edu> wrote: > >>Eckard Blumschein wrote: >>> On 12/5/2006 2:13 PM, Bob Kolker wrote: >>> > For the latest time. Uncountability is a property of sets, not >>> > individual numbers. >>> >>> I know this widespread view. >> >>So you claim. However, last time I asked you to give the standard >>definitions, you failed. Care to try again? Define "countable" and >>"uncountable". > > Since according to you "definitions are only abbreviations" how about > def(countable)="Y" and def(uncountable)="Z"? > > ~v~~ David expects from me a confession of faith. Sorry. |