Cantor Confusion
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From: David Marcus on 26 Oct 2006 15:18 mueckenh(a)rz.fh-augsburg.de wrote: > I guess I can see now the reason of your problems: You don't > understand because you are looking for a tanslation always. There is no > translation required. The tree is that mathematics which deserves this > name. It is outside of your model, independent of ZFC, but generally > valid and, therefore; covering ZFC too. Thank you for finally admitting that your argument can't be given within ZFC. -- David Marcus
From: David Marcus on 26 Oct 2006 15:21 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > Ah, but none of them actually get the same conclusion you do. So, you > > and they can't both be right. > > There are several mathematicians understanding my argument, but the > conclusions are no in question here. Some have understood my argument > but as it does not fit their expectation and conviction over many > years, they try to find a fault. That is a legitimate process of > finding the truth by discussion. Even Virgil has understood, although > he does not try to find any error but plainly refuses the result > because there is a contrary proof by Cantor. And I know from many of my > students that they understood me, although they do not study > mathematics but various other topics. Please name the mathematicians that agree that your argument is correct. (Han doesn't count, since he says he is a physicist.) -- David Marcus
From: William Hughes on 26 Oct 2006 15:23 mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > Within the *real* numbers the limit does exist. And a decimal number is > > nothing more nor less than a representative of an equivalence classes. > > So we are agian at this point: The real numbers do exist. Not according to you (at least not by the usual definition). According to your reasoning, there can only be a finite number of real numbers ever defined. This makes limits kind of weird, since you cannot get arbitrarily close. > > For the real > numbers we have LIM 10^(-n) = 0. Therefore, i ther limit n--> oo tere > is no application of Cantor's argument. > Given that there are only a finite number of integers, what do you mean by n--> oo ? (Or is the isomophic subset of the real numbers actually bigger than the integers?). You really need to have the courage of your own convictions. - William Hughes
From: Lester Zick on 26 Oct 2006 15:29 On Wed, 25 Oct 2006 18:12:03 -0400, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >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. Or you could try reading what I actually wrote. Your choice. ~v~~
From: Virgil on 26 Oct 2006 15:31
In article <1161883626.539177.12550(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > 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. > > According to set theorists, naming it is having it. That is another instance of "Mueckenh"'s profound misunderstanding of things mathematical. Naming something in mathematics no more requires that it exist in mathematics than naming Santa Claus causes him to exist in the physical world. >At least the > natural numbers and other infinite sets are present by naming N etc. > How else should an infinite set come to existence in any primordial > model without an axiom of infinity securing its existence? The axiom of infinity does not name anything, it declares the existence of something without actually naming anything. > > > > 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. > > > I know that. But some "experts" here are of the opinion that even ZF > without INF is an infinite theory. Does ZF without an AoI explicitily prohibit infinite sets, or merely no longer explicitly declare them to exist. Without an explicit prohibition, there will be models of (ZF - AoI) which contain infinite sets. > And when I consider the abuse of > expressions in set theory Compared to WM's list of abuses of expressions, ZF is pristine. > I cannot but believe, that everything is > possible there, even without the primordial existence of the set of all > sets there: There we have the actual, i.e., finished infinity, the > well-order of sets which definitely cannot be defined and, hence, > cannot be well ordered, the uncountability of countable sets, ... Why > then not the infinity of finite theories and sets? Since WM chooses to wallow in ignorance about what ZF, et al, actually say, he will also remain in ignorance of all the why's and why not's. |