Cantor Confusion
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From: David Marcus on 16 Nov 2006 01:48 Lester Zick wrote: > On Sat, 11 Nov 2006 13:50:38 -0500, David Marcus > <DavidMarcus(a)alumdotmit.edu> wrote: > >stephen(a)nomail.com wrote: > >> I think a lot of this "opposition" would go away if the word > >> "transfinite" instead of "infinite" had been used to describe > >> a set that can be put into a one-to-one correspondence with > >> a proper subset of itself. The word "infinite" sends people > >> down strange philosophical paths, as does the word "infinity" > >> despite the fact that it is not really even used in set theory. > >> Noone would argue about "transfinity". > > > >You could be right. Although, it seems unfair of the cranks to dictate > >what words mathematicians can appropriate. It is hard to make up good > >names. We have enough names like "second category" as it is. > > Once more we have this sloppy word usage on the part of those who self > righteously proclaim their mathematical rectitude. What exactly does > "crank" mean besides "crank(x)=disagree(u)"? Modern mathematikers > routinely appropriate words and make up private definitions for them > as if they were the sole arbiters of truth in mathematical terms. Of > course it's hard to make up good names especially when mathematikers > insist on private definitions cast in parochial terms of modern math. "Crank" means someone who makes up pejorative names for people whose language they do not understand and thinks that definitions which are explained in many books are "private". -- David Marcus
From: David Marcus on 16 Nov 2006 02:02 Lester Zick wrote: > On Sat, 11 Nov 2006 15:29:18 -0500, David Marcus > <DavidMarcus(a)alumdotmit.edu> wrote: > > >Lester Zick wrote: > >> Let me see if I can simplify how the issue is or ought to be argued. > >> You posit certain properties of an infinite however you define it. So > >> the question then becomes whether your claim is or can be true. > > > >What does "is or can be true" mean? In mathematics, we are normally only > >concerned with provability (unless discussing philosphy). > > "Provability" of what pray tell? If you're not concerned with proving > the truth of what you say in mathematics exactly when are you not > discussing philosophy every time you say anything in mathematics? Do you really not know the mathematical meaning of the word "prove"? If so, I (and others) could try to explain it to you. But, if you are just being argumentative, we won't bother. > > He seems to be saying that > >the notion of a completed infinity leads to either absurdities or > >contradictions. > > Well as far as I can tell it does. Most notably the containment of > sets and subsets. I don't remember as the calculus ever requires > infinites. Please give a specific example of something that you think is absurd or a contradiction. I don't know what you mean by "containment of sets and subsets". As to whether calculus requires infinities, it depends on what you mean by that. You can certainly do beginning calculus without using cardinality of infinite sets. The calculus was invented long before set theory was. > > Perhaps he thinks the way to avoid these absurdities is > >to only consider things that can be physically produced. > > Well that would certainly be one way. Another would be to stop using > finite infinites. Infinites only make sense in relation to one another > and not in relation to finites. Sorry. I don't know that you mean. What's a "finite infinity"? > >> Now personally I find most of the arguments disingenuous on both > >> sides. > > > >what is a standard mathematical argument that you find disingenuous? > > Insistence on the rectitude of arguments which can't be demonstrated > true. WM's arguments seem to rely on a finite universe. Opposition to > his arguments seem to rely on problematic assumptons of set theory. > Niether seems very convincing yet adherents of each pretend they are. I don't think anything in mathematics is "demonstrated true" in the sense you mean, although I'm not really sure what you mean. Are you serious that you can't tell the difference between WM's ravings and what other people are saying? WM is a classic crank. He insists on using words in nonstandard ways, refuses to explain what his words mean, claims that standard mathematics is absurd and/or contradictory, yet is clearly unfamiliar with basic mathematics. The people who bother to banter with him are saying things that are explained in many textbooks. You can read any of these textbooks and check for yourself that the proofs are correct. Mathematics may be unique in that the student can check everything themselves. There is no need to take anyone's word for anything. -- David Marcus
From: David Marcus on 16 Nov 2006 02:28 Lester Zick wrote: > On Sat, 11 Nov 2006 15:53:40 -0500, David Marcus > <DavidMarcus(a)alumdotmit.edu> wrote: > > >Lester Zick wrote: > >> On Fri, 10 Nov 2006 18:19:05 -0500, David Marcus > >> <DavidMarcus(a)alumdotmit.edu> wrote: > >> > >> >I think he has a bigger problem. He doesn't seem to agree that there are > >> >infinite sets. It is very strange. > >> > >> You mean if the editorial "we" agree that there are infinite sets > >> there are infinite sets? > > > >I don't know what you mean. What does "there are" mean in this context? > > You used the phrase and you're asking me? I know what I meant, but my meaning doesn't seem to make sense if I use it in your sentence. > I can't really say what you > meant when you used it. I mean capable of demonstration of truth. Sorry. I don't know what "demonstration of truth" means. Can you give an example of demonstrating the truth of something? > All > I've seen people do in connection with infinites is assume certain > properties and characteristics for infinites which they can't then > demonstrate are actually true. Showing an entity which had such > properties and characteristics would certainly be one way to prove > their truth and show that "there are" infinites. The only alternative > would seem to be some other form of demonstration. In neither case > would the mere hypothetical assumption of truth demonstrate anything. Not sure what you mean. In mathematics, all we do is pick some axioms, then see what theorems we can prove. Of course, we think the resulting systems are useful and/or interesting, but that doesn't "prove their truth". The usefulness to other fields is demonstrated via the scientific method, not by mathematical proof. > >> You have a very curious sense of words in > >> others but not in yourself. You claim to be able to prove things > >> without being able to prove they're true. > > > >I'm using "prove" in its mathematical sense. I don't know what you mean > >by "prove they're true". I suspect the meaning of the word "prove" is > >different in the two senses. > > No doubt that's true. The problem I have is that every time you use a > word, we have no idea whether what you're saying is supposed to be > true or philosophical. So what if you "prove" something mathematically? > Is the thing proven necessarily and universally true in mechanically > exhaustive terms? If not it's a systematic philosophical exposition at > best. That's why mathematikers coin definitions in ambiguous parochial > terms that would embarrass a sixth grade school teacher. Don't know what you mean. Are you saying you don't know what the word "proof" means in mathematics? > >> And what if one doesn't agree that there are infinite sets? > > > >If you mean you want to use different axioms for your mathematics, then > >you are welcome to. It that's not what you mean, then I don't know what > >you mean. What does "there are" mean in your sentence? > > I don't want to use any axioms for mathematics. That's the point. > That's what got math into the pickle it's in with all its parochial > axiomatic assumptions of truth and private ambiguous definitions. How can you do mathematics without axioms? A major purpose of axioms is to avoid ambiguity. > >> Are you going to prove they're true? > > > >I don't understand the question. > > Are you going to illustrate the existence of infinites by production > of one or more; or are you going to demonstrate the truth of their > existence by some alternative means? You posit certain properties and > characteristics of things you call "infinites" but don't show they can > actually be realized in combination with one another. Sorry. Don't know what you mean. In particular, I don't know what you mean by "illustrate the existence", "demonstrate the truth of their existence", "actually be realized". Can you give an example? All we do in mathematics is prove theorems from axioms. That's it. > >I take words seriously enough to be sure that I and the person I am > >conversing with are using the words with the same meaning before I jump > >to any conclusions. > > You jump to every conclusion like a twelve year old boy jacking off. > How is it you verify you are using words with the same meaning if you > can't demonstrate the truth of what you're saying in mechanically > exhaustive terms capable of comprehension by others in identical > terms? That's what mathematical formalisms are for. But just saying > they're mathematical formalisms doesn't necessarily make them true and > doesn't make them mechanically reducible in exhaustive terms. I can verify I'm using words with the same meanings as other people by asking them what definitions they are using, then seeing if they are the same as mine. Similarly, I can read a book, and see what definitions the book uses. Have you read any math books at the junior/senior college level or above? > I'll tell you exactly what you do. You assume you "know" what you're > talking about because you've spent many years studying the arcana you > call mathematics in terms identical with others who have done the > same. Then you have no difficulty at all conversing with those others. Exactly! Of course, anyone is welcome to read the books and do the same. > But when someone comes along who wants to question the truth of the > paradigm itself you accuse him of not doing mathematics and call him a > crank. No. The people who tell us that what we are doing is wrong, but who don't understand the rules of the game we are playing, are the ones who are cranks. > Now I don't say there aren't cranks out there but there is also > truth out there and you don't have a clue as to how to get at it. Why do you think "truth" is relevant to mathematics? > > Mathematics is a language. People who learn the > >language communicate using that language. People who are not fluent in > >the language may misunderstand what is being said, since many of the > >same words are used as in English. However, the meaning of many words is > >different in the two languages. > > Of course mathematics is a language. The problem is that mathematics > is or should be an exhaustively true language and not merely a dialect > used by a certain group of professionals who choose to call themselves > mathematicians
From: David Marcus on 16 Nov 2006 02:31 William Hughes wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > omega, omega = 2omega > No > > You want to do is something like take > > 2omega is the ordinal of the sequence 1,3,5,...,2,4,6 ... > > and then substiute omega. But what you are saying is > > the ordinal of the sequence 1,3,5,...,2,4,6 ... is the same as the > ordinal of the > sequence omega,omega, which is nonsense. The sequence omega,omega > has an ordinal of 2. > > > 1,3,5,..., omega, 2,4,6,...,omega = 2omega + 1 > > This is true (it is of course not the only sequence that > represents the ordinal 2omega +1) However, even using your nonsensical > formal manipulations we get > > 1,3,5,..., omega, 2,4,6,...,omega = 2omega + 1 > (1,3,5,...),(omega,2,4,6,...),omega = 2omega +1 > omega,omega,omega = 2omega+1 > > Which does not contradict anything you wrote. Which nonsensical > formal manipulation were you thinking of? What a marvelous question! -- David Marcus
From: Eckard Blumschein on 16 Nov 2006 02:39
On 11/16/2006 2:17 AM, Dik T. Winter wrote: What are the things that represent numbers? My humble trial to answer this question is: Infinite trees represent real numbers. All nodes represent rational numbers. So it is perhaps impossible to perform a quantitative comparison. Galilei Galilei was not arrested in vain. Eckard Blumschein |