Cantor Confusion
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From: Eckard Blumschein on 5 Dec 2006 11:11 On 12/4/2006 10:02 PM, Bob Kolker wrote: > Eckard Blumschein wrote: >> >> For my feeling, Dedekind and Cantor were lacking power of abstraction. > > Actually they had more such power than Kronecker was comfortable with. > > Bob Kolker Kronecker failed to have support by close friends. He was too aristrocratic, overly self-confident and perhaps mentally not robust enough. Dedekind, Cantor, Heine, Hurwitz, Weierstrass, Frege, Russell, Peano, Hilbert, Zermelo, Bernstein, Mittag-Loeffler, du Bois-Raymond, Hausdorff, Hessenberg, Lebesgue, Schoenflies, Fraenkel, and many others mutually complemented and supported each other.
From: Eckard Blumschein on 5 Dec 2006 11:31 On 12/4/2006 9:23 PM, MoeBlee wrote: > Eckard Blumschein wrote: >> Correct. There are people who extend the reals to include oo. > > Would you give an example of a text that does this? > > What we sometimes do is add two points (called 'oo' and '-oo') to the > real number system so that we have a different, extended system (which > is not a complete ordered field). But that does not meant that we > consider oo and -oo to be real numbers. > > MoeBlee > I already replied.
From: Eckard Blumschein on 5 Dec 2006 11:59 On 12/4/2006 8:47 PM, Virgil wrote: > In article <1165238765.397374.303270(a)79g2000cws.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: >> Most "mathematicians" even don't know what potentially infinite is. > > As it is a useless idea, such ignorance is bliss. And WM's sinful > attempts to destroy that innocence is reprehensible. Cantor still understood that the Aristorelian potentially infinite point of view is quite different from actual infinity. The formerly Archimedean axiom of infinity describes the potential infinity. Blissful ignorance of mathematicians does not utter complains if the axiom of (possibly infinite) extensionality claims the existence of a set which has to include all of its elements. According to my reasoning this does neither clearly include nor clearly exclude the actual infinity, i.e. all elements together. Nobody complains. Obviously, the fiction of actual infinity is merely required from theoretical point of view. Nobody really needs it in practice. This preserved ambiguity lead to the theoretical imperfections I reported.
From: Eckard Blumschein on 5 Dec 2006 12:56 On 12/5/2006 1:26 PM, Bob Kolker wrote: > Eckard Blumschein wrote: > >> Obviously, you meant not either but neither? >> >> I wonder why the mathematicians believe to require one-point >> compactification. I consider the rationals as genuine numbers, being as >> close as you like to the fictions infinity and real numbers. The exact >> numerical representation of pi requires the fiction of actual infinity. >> > Rational numbers are non-genuine. Nowhere in the physical world outside > of our nervouse systems do they exist. Genuine does not refer to physics but 1) to the possibility to be derived from natural numbers by a finite number of steps 2) to being discrete in the sense to have a distinct numerical address 3) being consequently countable > For the same reason, integers are non-genuine. > > The only genuine things in the cosmos are physical. I did not spoke of genuine things in the cosmos. > > Bob Kolker
From: Eckard Blumschein on 5 Dec 2006 13:02
On 12/5/2006 2:13 PM, Bob Kolker wrote: > Eckard Blumschein wrote: >> an exact numerical representation available. Kronecker said, they are no >> numbers at all. Since the properties of the reals have to be the same as >> these of the irrationals, all reals must necessarily also be uncountable >> fictions. > > For the latest time. Uncountability is a property of sets, not > individual numbers. I know this widespread view. > There is no such thing as an uncountable real > number. Real numbers according to DA2 are uncountable altogether. People like you will not grasp that. Not a single real number is countable. > Nor is there any such thing as a countable integer. Every single integer is a countable element. > Countability > /Uncountability are properties of -sets-, not individuals. Do not reiterate what I know but deny. > > You have been told this on several occassions and you apparently are too > stupid to learn it. You should try and refute. Good luck. > > Bob Kolker |