Cantor Confusion
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From: Virgil on 31 Oct 2006 16:36 In article <1162298882.038772.64700(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1162157745.995131.292260(a)k70g2000cwa.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Dik T. Winter schrieb: > > > > > > > "positive numbers" with "numbers larger than 0". Because I understand > > > > the > > > > main domain is Anglo-Saxon mathematics. This is in contradiction to > > > > what > > > > I did learn at university (0 is both positive and negative). > > > > > > Really? I never heard of that. Is here anybody who learned that too? It > > > would interest me. No polemic intended. > > > > > > Regards, WM > > > > French mathematical usage, particularly Bourbaki, uses "positive" > > and "strictly positive" where the correspponding English mathematical > > usage is, respectively, "non-negative" and "positive". Similarly for > > "negative' and "strictly negative" versus "non-positive" and "negative. > > > > Thanks, it is the same in German. WM being ambiguous again. Is the German usage the same as English or the same as French? > > Regards, WM
From: Virgil on 31 Oct 2006 16:37 In article <1162299073.515449.110270(a)e64g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > > > Do and enjoy your mathematics. I will not disturb you. > > > > Then why are you posting to sci.math? What do you possibly hope to > > achieve? > > To help those who have not yet decided to study set theory to avoid > wasting their time. > > Regards, WM So that they can waste it listening to your foolishness instead? Hardly a bargain.
From: Virgil on 31 Oct 2006 16:43 In article <1162299188.165885.80340(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > Dik T. Winter schrieb: > > > > > > > Euclides was apparently smarter than Cantor in this. He used the > > > > parallel axiom (postulate) for something he could not prove from the > > > > other axioms. > > > > > > But which is obviously possible and correct under special > > > circumstances, while Zermelo's AC is obviously false under any > > > circumstances. > > > > What do you mean an axiom is "false"? > > An axiom leading to a contradiction when added to a set of axiom which > do not so, is false. The axiom that every straight line crosses itself > is false in Euclidian geometry. A statement does not become an axiom as a part of some axiom system, until it is assumed to be true in that system. A statement whose negation is provable in any system is unlikely to become assumed true in that system, and therefore is unlikely to become an axiom, at least in that system. The statement that every straight line crosses itself is false in Euclidean geometry, but I have never heard of it being nominated for axiomhood in Euclidean geometry. > > > Are you discussing philosophy or > > mathematics? > > I don't know what predominates in Cantor's theory - philosophy or > theology, but I am sure that there is no mathematics there. Wm claims to be certain of a lot of things that are not so.
From: Virgil on 31 Oct 2006 16:52 In article <1162299524.423928.41670(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > The definition is sufficiently special (= non-general) to determine > > > that lim [n-->oo] {1,2,3,...,n} = N is correct. > > > > Yes, of course, because that was the definition you provided. So because > > it is so defined, the definition is correct. > > > > > > > > The operator "lim [n-->oo]" defines N. In your example lim {n -> oo} > > > {-1, 0, 1, ..., n} we have N too but in addition the numbers -1 and 0. > > > > "lim {n -> oo}" is an operator that works on sequences, apparently. And > > you have defined that operator for precisely one sequence of sets. > > It is defined for infinitely many sets of integers. > > lim [n-->oo] {-k,-k+1,..., 0, 1,2,3,...,n} = {-k,-k+1, ...,0} u N for > every k e N. > lim [n-->oo] {k, k+1, k+2,...,n} = N \ {1,2,3,...,k-1} for every k e N. These are, at best, operational definitions which hold in particular cases but do not define any general rule. Given, say, f:N -> P(N), a function from N to the power set of N, Under what conditions on f can one say that there is an S in P(N) such that lim[n -> oo] f(n) = S ? Until WM has answered this question, or something quite similar, he has not defined what his limiting process is, nor how to test his claims for validity. > Don't you see that the whole aim of Newspeak is to narrow the range of > thought? In the end we shall make thoughtcrime literally impossible, > because there will be no words in which to express it. (George Orwell > in "1984") Which is why we avoid using WM's version of Newspeak, as well as Orwell's. Mathspeak is quite different from both WMspeak and Newspeak.
From: stephen on 31 Oct 2006 16:53
imaginatorium(a)despammed.com wrote: > David Marcus wrote: > <snip> >> I wonder: do Lester, Ross, Han, Tony, and WM all agree that noon doesn't >> exist? It is such an odd thing to say. Imagine walking up to someone in >> the street and trying to convince them that noon doesn't exist. >> >> It is so hard to keep the nonsense straight. It all seems to run >> together--although there are stylistic differences. > More than stylistic, I think. For example, don't offhand recall Lester > ever saying anything mathematical that was even wrong, Oh Lester has definitely said things that were coherent enough to be wrong, as opposed to just being incomprehensible. For example he insists that dr/dt = r/t where r is the radius of a circle and t is time. Stephen |