Cantor Confusion
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From: Lester Zick on 25 Oct 2006 18:04 On Wed, 25 Oct 2006 21:01:08 +0000 (UTC), Sebastian Holzmann <SHolzmann(a)gmx.de> wrote: >mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: >> Here I repeat some some education for you (by Fraenkel, that is one of >> those scholars who made ZF): Vor allem ist die Bildung der wichtigsten >> Klasse paradoxer Mengen, n?mlich der allzu umfassenden Mengen >> (Antinomien von Burali-Forti, Russell usw.) durch unsere Axiome >> ausgeschlossen. Denn diese gestatten, eine oder mehrere gegebene Mengen >> als Ausgangspunkt nehmend, nur entweder die Bildung beschr?nkterer >> Mengen durch Aussonderung bzw. Auswahl, oder die Bildung von Mengen, >> die in eng umschriebenem Ma? sozusagen umfassender sind, durch >> Paarung, Vereinigung, Potenzierung usw. > >Please do not quote scholars from some remote time on me. Rest assured >that I try not to sound as foolish as you do. But perhaps you succeed without trying. ~v~~
From: Lester Zick on 25 Oct 2006 18:07 On 25 Oct 2006 13:14:22 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >mueckenh(a)rz.fh-augsburg.de wrote: >> Sebastian Holzmann schrieb: >> >> >> > Oh, I think I begin to see your problem here. But before we can speak of >> > ZFC as a theory, we must first have some sort of "background set theory" >> > available. And if we do not allow that background theory to "have" >> > infinite sets (in some naive way), we cannot even formulate Z, because >> > it consist of infinitely many sentences... >> >> So in principle we need the set of all sets in order to talk about >> every thing including > >Who said anytying about talking about "every thing"? > >> the fact that it does not exist > >No, we don't need to hold that there exists a set of all sets to prove, >in a given theory, that there does not exist a set of all sets. > >> and that we >> cannot have any infinity unless we have the axiom of infinity. > >To prove there exists an infinite set requires some axioms. If the >axiom of infinity is not one of them, then, to prove the existence of >an infinite, there has to be some other axiom or axioms aside from the >axioms of Z without the axiom of infinity. Well if by "axioms" you mean assumptions of truth I would certainly agree. ~v~~
From: David Marcus on 25 Oct 2006 18:12 Lester Zick wrote: > On Wed, 25 Oct 2006 00:44:47 -0400, David Marcus > <DavidMarcus(a)alumdotmit.edu> wrote: > >Lester Zick wrote: > >> On Mon, 23 Oct 2006 22:06:05 -0400, David Marcus > >> <DavidMarcus(a)alumdotmit.edu> wrote: > >> >Han de Bruijn wrote: > >> >> And no, binary trees will not be found in Halmos' "Naive Set Theory". > >> >> Because it's too naive, I suppose .. > >> > > >> >I don't see how you would know it is not there since you said that while > >> >you own the book, you've never read it. Funny how books aren't much use > >> >if you don't actually read them. > >> > > >> >Anyway, I didn't say binary trees were in the book. I said that if > >> >Muckenheim wants to talk using the mathematical concepts of sets, > >> >functions, and relations, he should use standard terminology, where > >> >"standard" means the same as in some mathematics book. It is like saying > >> >that if he wants to speak in English he should use words according to > >> >the meanings as given in an English dictionary. > >> > >> So you're suggesting we should argue against a paradigm in terms set > >> by the paradigm? So perhaps I should argue my definition of "infinity" > >> in standard set analytic terms even though cast in Newtonian terms of > >> the calculus? Curious to say the least. > > > >You didn't actually read what I wrote, did you? > > I'm asking for clarification of what you wrote. > > >Mueckenheim claims his example is part of standard mathematics. That's > >the whole point, since he claims it shows standard mathematics is > >inconsistent. If the example can't be given in standard mathematics, > >then it can't show that standard mathematics is inconsistent. > > > >If the example is part of standard mathematics, Mueckenheim should be > >able to state it using standard terminology. Of course, he is free to > >use any new terminology he wishes, as long as he identifies it as such > >and defines it. > > > >If Mueckenheim would merely admit that he was creating a new type of > >mathematics, most people would leave him to it. > > If we modify the standard mathematical analytical paradigm we're > scarcely doing standard mathematical analysis. I have no idea what > Muekenheim's concepts are or what innovations he intends. But your > claims seem to portend that no one can argue against the standard > mathematical analytical paradigm except in terms of the standard > mathematical analytical paradigm, which I find absurd. I know lots of > theologians who argue the same way: you can't criticize what you don't > understand and you can't understand what you haven't studied in detail > and there'll always be someone out there who has studied more than you > so you can never effectively criticize them and their opinions at all. > The paradigm itself can be wrong just as definitions can be false if > based on false propositions and combinations of predicates. And that > wouldn't make it a new type of mathematics it would just make it a > correct type of mathematics. Try reading what I actually wrote. -- David Marcus
From: David Marcus on 25 Oct 2006 18:20 mueckenh(a)rz.fh-augsburg.de wrote: > > Han de Bruijn schrieb: > > > David Marcus wrote: > > > > > Han de Bruijn wrote: > > > > > >>And no, binary trees will not be found in Halmos' "Naive Set Theory". > > >>Because it's too naive, I suppose .. > > > > > > I don't see how you would know it is not there since you said that while > > > you own the book, you've never read it. Funny how books aren't much use > > > if you don't actually read them. > > > > > > Anyway, I didn't say binary trees were in the book. I said that if > > > Muckenheim wants to talk using the mathematical concepts of sets, > > > functions, and relations, he should use standard terminology, where > > > "standard" means the same as in some mathematics book. > > There are many books on binary trees and many others on geometric > series. I put these topics together. Randomly mixing terms from two different fields into a sentence does not make the sentence meaningful. > That is new and not everybody can > understand it immediately. You must not be sad on that behalf. I'm only sad that you really seem to believe what you say. > Follow > just my discussion with those guys who have understood this > comparatively simple matter. Ah, but none of them actually get the same conclusion you do. So, you and they can't both be right. > At the end, you may get it too. > > > > Or, he could continue to do as Humpty Dumpty did and use his own > > > meanings for words without telling people what they are. > > You must not think that everything you cannot understand is > ununderstandable. It is just some new idea. Or, it is not an idea at all, but just random words. > Otherwise it would not be > so interesting. You prefer to learn old knowledge from your old books > with their old trodden paths? My paths are new! Kind of ironic considering you haven't read anything published since the early 1900's. -- David Marcus
From: David Marcus on 25 Oct 2006 18:23
mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > In article <1161683454.085728.22020(a)m73g2000cwd.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > Therefore we write the limit. The limit of all n is omega. > > > > By what definition of "limit"? > > > > > But the *facts* are based on the number of transactions. The trick with > > > the vase is simply to force the infinite number of transaction into a > > > finite time interval. > > > > The only relevant question is "According to the rules set up in the > > vase problem, is each ball which is inserted into the vase before noon > > also removed from the vase before noon?" > > > > An affirmative answer confirms that the vase is empty at noon. > > A negative answer directly violates the conditions of the problem. > > > > How does Mueckenheim answer? > > You see my name written above. My first name is Wolfgang, my short sign > is WM. Unless you switch to one of these alternatives, this is my last > answer to you - be sure. > > All the balls have been removed before noon. OK. > But more balls are in the vase. Reason? Proof? Example? Anything? > Conclusion: It is nonsense to talk about the complete set N and about > any infinite set, no matter, whether it is done in English or in any > formal language. -- David Marcus |