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From: Will Twentyman on 7 Apr 2005 18:03 > > I dealt not just with such mediocre proponents of a wrong concept like > Zermelo and the remedies they introduced but also with Hilbert. You will > find a few comments of mine at > > http://iesk.et.uni-magdeburg.de/~blumsche/M280.html > Reading this, it seems that your objection is that Cantor's work does not correspond to your notions of infinity. So be it, just worry about whether or not there are bijective maps between sets, set up equivalence classes for those that do have such maps, and order the equivalence classes based on injective/surjective maps existing. Don't worry about "infinity". -- Will Twentyman email: wtwentyman at copper dot net
From: Eckard Blumschein on 8 Apr 2005 03:17 On 4/8/2005 12:03 AM, Will Twentyman wrote: >> >> I dealt not just with such mediocre proponents of a wrong concept like >> Zermelo and the remedies they introduced but also with Hilbert. You will >> find a few comments of mine at >> >> http://iesk.et.uni-magdeburg.de/~blumsche/M280.html >> > > Reading this, it seems that your objection is that Cantor's work does > not correspond to your notions of infinity. Admittedly, I am not aware of any other reasonable notion of infinity than expressed by Aristotele: Infinitum actu non datur, by Spinoza: Infinity cannot be enlarged, by Gauss: "so protestiere ich ... gegen den Gebrauch einer unendlichen Grýýe als einer Vollendeten, welche in der Mathematik niemals erlaubt ist" and by all other serious (i.e. non Cantorian) mathematical literature. Those, in particular Galilei and Leibniz, who did not yet entirely agree on how to calculate with infinity, nonetheless fully agreed on some basic rules like oo+a=oo and oo^oo=oo. Everybody except for Cantor's fellows understands why we write x->oo but not x=oo. I am trying to explain that infinity is correspondingly just a quality, not a quantity. There are two questions: 1) Why did Cantor ignore the only reasonable notion of infinity including the pertaining rules how to handle it? Presumably he articulated widespread confusion between "infinite" and "very large". I noticed that even Poisson and Weierstrass used the nonsense term "infinite number". Most likely Cantor was driven by an insane ambition to create something new. Recall Dedekind's careless opinion that in mathematics any creation is allowed. 2) Why did he manage to find so much support? I would like to abstain from answering this question because it relates to human fallibility rather than mathematics. Warnings by Kronecker, Poincarý and many others were ignored from Weierstrass and others. The same Bertrand Russell who declared causality a relic of bygone time like monarchy and who suggested a preventive nuclear stroke that would have killed me wrote: "The solution of the difficulties which formerly surrounded the mathematical infinite is probably the greatest achievement of which our age has to boast". Well, boast is the matching word if one tries to judge how demagogically Hilbert, Fraenkel and others promoted Cantor's transfinite numbers. The weaker the argument, the stronger the euphoria. > So be it, just worry about > whether or not there are bijective maps between sets, I cannot see any reason for that. Cantor was correct in that rational numbers are countable in the sense, they can be brought into a one-to-one relationship to the infinite set of natural numbers. Please notice, I am perhaps the first one who does not try to refute Cantor's evidence. This bings me to a fresh idea. There might ideed be a mathematical aspect answering in part question 2): Cantor was like a matador. In particular his second diagonal argument porovoked a lot of failed attempts for refutation making him more and more famous. I realized that the argument might be formally correct. All the attackes felt that Cantor was wrong. Being, however, not aware of the true location of Cantor's fallacy, they made the same mistake like Cantor himself. Please read M280. A translation into German is available elsewhere and might hopefully a little bit more understandable in some decisive details. Having originally stated that there are not more real than rational numbers, I tried to exclude the possiblity to be mistaken by writing: "There are neither more nor less nor equally many real numbers as compared to the rational ones." Eckard > set up equivalence > classes for those that do have such maps, and order the equivalence > classes based on injective/surjective maps existing. Don't worry about > "infinity".
From: Robert Kolker on 8 Apr 2005 04:52 Eckard Blumschein wrote: > himself. > Please read M280. A translation into German is available elsewhere and > might hopefully a little bit more understandable in some decisive > details. Having originally stated that there are not more real than > rational numbers, I tried to exclude the possiblity to be mistaken by > writing: "There are neither more nor less nor equally many real numbers > as compared to the rational ones." I assume that you never never ever use the Axiom of Choice. Is that the case? Bob Kolker
From: Eckard Blumschein on 8 Apr 2005 05:48 On 4/8/2005 10:52 AM, Robert Kolker wrote: > Eckard Blumschein wrote: > >> himself. >> Please read M280. A translation into German is available elsewhere and >> might hopefully a little bit more understandable in some decisive >> details. Having originally stated that there are not more real than >> rational numbers, I tried to exclude the possiblity to be mistaken by >> writing: "There are neither more nor less nor equally many real numbers >> as compared to the rational ones." > > I assume that you never never ever use the Axiom of Choice. Is that the > case? Of course. I am not a mathematician, and as far as I know not even all mathematicians prefer to use AC, e.g. Bell, while perhaps most physicians prefer ZFC against ZF. My pertaining knowledge is very limited. However, I looked into Cantor's work in order to understand how he imagined to make sure that a set can be made well-ordered. His concern was the first element of a set. Well, when Julius Koenig in 1904 objected against the possibility of a well-ordered set of the reals, Ernst Zermelo came up with the remedy of AC. Following Dedekind's credo that mathematics is to be created arbitrarily, one may add as many most artificial axioms as one likes provided they do not contradict each other. Maybe, I am to sensitive as to behave this way. According to my reasoning, definition of the first element does not at all make the reals a well-ordered set. The real problem is the impossiblitity to numerically distinguish any real number from its immediate successor. I consider a choice function inside the reals a self-requiring assumption. Eckard Blumschein
From: Dave Rusin on 8 Apr 2005 10:24
Eckard Blumschein wrote: > >> "There are neither more nor less nor equally many real numbers >> as compared to the rational ones." Did you mean to say the _number of_ real numbers is the same as the _number of_ rational numbers? There are some fellows over in comp.ai.philosophy and sci.philosophy.meta you should talk to. dave |