Cantor Confusion
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From: MoeBlee on 2 Nov 2006 16:04 mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1162405520.008395.100850(a)e64g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > ... > > > > Finally you give a definition. Why did it take so long? > > > > > > I thought that this was so clear that no explanation was required. > > > > I asked you for a definition quite a few times. I would have thought > > that that was enough indication that it was not clear at all. > > > > > > > > And in mathematics 1/oo is *not* defined. > > > > > > > > > > Not in mathematics. But in a theory which assumes omega to be a whole > > > > > number. > > > > > > > > Which theory? > > > > > > Set theory. > > > Cantor invented omega and defined omega as a whole number. > > > Who changed this standard meaning? > > > Why do you think this meaning was changed? > > > When do you think the contrary meaning became standard? > > > What is the contrary meaning? > > > Do you agree that A n: n < omega is incorrect? > > > If not, why do you complain about on-standard meaning on Cantor's > > > definition? > > > > A nice rant. > > Why don't you answer? > > > Where did Cantor define 1/oo? A quote might be > > appropriate. > > see below > > > > > > Where in the above quote is 1/oo defined? > > > > > > It is defined that omega (which Cantor used later instead of oo) is a > > > number larger than any natural number n. Omega is the limit ordinal > > > number. Therefore 1/omega must be a number smaller than every fraction > > > 1/n. > > > > Why? As long as 1/omega is not defined you can not talk about it. You > > simply assume that 1/omega is a number. But that is not the case, it > > is not defined. > > If omega is a number > n, then 1/omega is a number < 1/n. For all n e > N. > > > > > > > Therefore we have there limit ordinal numbers? > > > > > > > > Yes. So what? Limits are in general not defined. > > > > > > In particular the limit of all segments of N is not defined. But set > > > theory does it. > > > > No. And if so, show me where. > > Es ist sogar erlaubt, sich die neugeschaffene Zahl omega als Grenze zu > denken, welcher die Zahlen n zustreben, wenn darunter nichts anderes > verstanden wird, als daß omega die erste ganze Zahl sein soll, welche > auf alle Zahlen n folgt, d. h. größer zu nennen ist als jede der > Zahlen n. (p. 195) > > This definition has been conserved up to our days: Limesordinalzahl or > limit ordinal number. I regret that I don't read German. But I'd like to know what you might propose as a mathematical definition of 1/omega in Z set theory. Notice, I'm not asking what Cantor wrote. I'm asking what is the definition of 1/omega specifically in Z set theory. MoeBlee
From: MoeBlee on 2 Nov 2006 16:15 mueck...(a)rz.fh-augsburg.de wrote: > My question is : Do you maintain omega > n for all n e N? I know that > modern set theory says so. If something can be larger than a number, > then it must be a number. You're again asking questions that show you have no business quoting Fraenkel, Bar-Hillel, and Levy as if you knew what they are talking about. For ordinals, '<' refers to a specific relation (class) among ordinals. For ordinals, x<y <-> xey where 'e' is the epsilon membership symbol. > If it cannot be a fraction because ZF does > not yet know how to divide elements, In ZF we define various operations of division. As far as I know, there is not a dvision operation for sets in general. > then it can only be a whole > number, I would guess.# These problems you're having are of fitting set theory to your own system of terminology. To work in set theory, we don't need to care about your own system of terminology. MoeBlee
From: Lester Zick on 2 Nov 2006 17:17 On Fri, 03 Nov 2006 02:27:20 +0800, Noehl <n.alinsangan(a)gmail.com> wrote: >Lester Zick wrote: >> On Thu, 2 Nov 2006 00:28:49 -0500, David Marcus >> <DavidMarcus(a)alumdotmit.edu> wrote: >> >>> Dik T. Winter wrote: >>>> In article <sjnfk256m5tau0bmk5u4vjdg53al1f2sc9(a)4ax.com> Lester Zick <dontbother(a)nowhere.net> writes: >>>> > Yours are some of the very >>>> > few that don't. And I've seen posts of other .nl correspondents which >>>> > come through in color. >>>> >>>> It has nothing to do with the domain where you come from. It has everything >>>> to do with the manner things are quoted and in the way *your* newsreader is >>>> able to handle that. I know the reason your newsreader does not show colours. >> >> But my newsreader does show colors. >> >>>> It is because I prepend the '>' sign with a space when quoting. >> >> Ok. Mine reacts the same way. But when there is no preceeding space >> quoted text is shown in a different color. >> >>> And I >>>> have pretty good reasons to do that. (One of the reasons being that this >>>> article would be rejected because there is more quotation than new text.) >> >> I've never experienced that. >> >>> My newsreader lets me post articles that have more quotation than new >>> text. >> >> ~v~~ > >Thunderbird quotes nicely Yeah, I'm not sure what the problem is if there is a problem. ~v~~
From: David Marcus on 2 Nov 2006 18:57 Virgil wrote: > In article <1162470874.593282.36250(a)b28g2000cwb.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > WM merely repeated his automatic error several more times here. > > WM claims that a list in which the nth listed element is a string of > length at least n characters cannot produce a diagonal of length > greater that any finite number of characters. > > His claim is trivially and obviously false, but he keeps repeating it ad > nauseam, as if by sufficient repetition of that lie , he can make the > truth go away. Not only does he keep repeating it, but he never even tries to justify it in any way. He is like a broken record. We ask him for his reason and he just repeats the same unjustified, erroneous claim. Kind of boring. -- David Marcus
From: David Marcus on 2 Nov 2006 19:00
Virgil wrote: > In article <1162471594.046811.299910(a)f16g2000cwb.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > > > Fine. Then stop claiming that you can prove ZFC is inconsistent. If you > > > > > don't like ZFC, that's your concern. But, that is completely different > > > > > from saying that you have a proof within ZFC of a contradiction. > > > > > > > > I have a proof which shows that the real numbers have less elements > > > > than a countable set. That is all. ZFC and what there can be formalized > > > > does not bother me at all. > > > > > > If the proof cannot be given in ZFC, then it is irrelevant to whether > > > ZFC is inconsistent. > > > > I cannot construct a horoscope. > > Your time would be better spent learning to do so. Good one. -- David Marcus |