Cantor Confusion
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From: Lester Zick on 13 Nov 2006 15:21 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 can't really say what you meant when you used it. I mean capable of demonstration of truth. 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. >> 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. >> 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. >> 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. >> Clearly like most trained in the modern mathematical arts >> you don't take words seriously enough to form critical thoughts. > >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'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. 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. 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. > 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 to the exclusion of others. That's what mathematics is supposed to be all about. It's a lingua franca of numerical concepts, relations, and processes used in science of all types. Those words you refer to as the same in generic languages are supposed to be refined so as to be true in universal terms and accessible to all on that basis. They aren't supposed to represent any kind of private language used to define a mathematical clique. Mathematikers routinely appropriate words from generic language, give them private definitions in completely ambiguous terms, and then insist that if they're not used that way they're not mathematical. Hell from what I can tell of set theory as practiced in conventional terms the whole thing is nothing more than a rank assemblage of axiomatic assumptions of truth used to justify not the analysis of sets but the paradigm for the study of sets in ambiguous terms cast by practitioners of the paradigm. Then they use the paradigm to excuse themselves from having to explain whether and why the paradigm is true by saying that you can't criticize what you don't understand and you can't understand what you haven't studied and here's an entry level text for dummies who presume to criticize mathematicians' views on mathematics and don't come back until you understand how terms are used by real mathematicians so it won't be a waste of their time. Nor is the problem confined to mathematics alone but permeates all theoretical sciences from the lowest level quantum physics to the highest scale cosmology. The common problem is that none of what empiricists assume to be true is demonstrably true in mechanically exhaustive terms but they just go on ta
From: Virgil on 13 Nov 2006 15:41 In article <1163425753.113699.136190(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > >Everyone is > > welcome to choose their own axioms. > > That's mathematics? If one chooses a set of axioms no one else is interested in it is pretty much useless mathematics, but still mathematics. The most productive mathematics comes, by and large, from those sets of axioms that the largest groups are in general agreement about. > > > > > > And, you haven't given a definition of the term. > > > > > > I thought you'd know transitivity: Take the expression "finshed entity" > > > where entity is a variable like "the set X" in set theory. Now replace > > > this variable by a fixed set like N, which in mathematics, is an > > > infinite process. This leads to "finished infinite process", > > > abbreviated by "finished infinity". Was this simple enough for you to > > > understand? > > > > Amusing, but that just shows how you can make up new terms. You still > > haven't provided a defintion. It appears you are saying that "finished > > infinite process" and "finished infinty" are synonyms. Fine. But, you > > haven't defined either one. Please define them. > > Would like to do. Please le me know which words are available in your > universe of discourse. Any definition of a "finished" anything which is based on the word "finished" itself, is circular at best. > > > > > > > Which mathematics do you allude to? > > > > > > > > That which is taught in school, explained in textbooks, published in > > > > journals, and discussed by some in this newsgroup. > > > > > > I teach at a school, I wrote textbooks (and my new book is in print), I > > > published in journals and I wrote in newsgroups. > > > > Irrelevant since that isn't what my sentence means in English. > > Could you explain what your sentence means? For instance: How many > schools and textbooks do you require? Do you claim to teach mathematics at your school? If so, at what level and with what content. If you do not, then whatever you do claim to teach is irrelevant. Are any of your textbooks primarily mathematics texts? If not, what is their primary subject matter, not that it does matter.
From: Virgil on 13 Nov 2006 15:50 In article <1163426009.651510.237050(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > > It appears that you are saying that the terms "infinite set" and > > "infinite number" are not meaningful. There are only "potentially > > infinite sets" and "actually infinite sets". Is that correct? > > > > Do you agree with the following statements? > > > > c. It is possible to have a potentially infinite set of numbers that > > does not contain an infinite number. > > It is difficult to answer this question, because the expression "set" > is occupied in modern mathematics by collections of elements which are > actually there (you don't know what that means, imagine just a set as > you know it). Such infinite sets do not exist. While infinite collections in any physical sense are not possible, why are imaginary infinities, such as sets of numbers must be, unimaginable? How is an imagined set with no members any more actual that an imagined set with every natural number as a member? To me they are both equally figments of the imagination. How is an imagined horse any more real that an imagined unicorn? > If an actually infinite set of numbers existed, and if neighbouring > elements had a fixed distance from each other, then the set must > contain an infinite number. I can more easily imagine an infinite set of finite naturals than one which must contian any infinite natural. I cannot even imagine what an infinite natural (as distinct from an infinite ordinal) would be like. For such fantastic creations one should appeal to such as TO.
From: Virgil on 13 Nov 2006 15:55 In article <1163428639.664610.309690(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > Every column has a fixed number omega of terms. You can add 1 further > > > term and you will get omega + 1 terms in every column. Every line has > > > less than omega terms. If you add 1 term to every line, the number of > > > terms in every line remains less than omega. What about the diagonal? > > > It has to satisfy both conditions, which is impossible. > > > > No. Adding one element to each line does not change > > the supremum of the lengths of the lines, so it does > > not change the length of the diagonal. > > The supremum is larger than any line, so it must be larger han the > diagonal which is confined by the lines. Maybe longer than yours but not longer than mine. > > > > The number of terms in each line, n, is less than omega. > > The number of columns is the supremum of the number of terms > > in each line. The number of columns is omega. > > If there is no line with omega terms, what does the supremum of the > columns consists of? That assumes, falsely, that the supremum of a set must be a member of the set. The supremum of the set of all finite ordinals is the first limit ordinal, which is not a finite ordinal. > > > The number of lines is omega. > > The number of columns is equal to the number of lines. > > You can repeat your prayer as often as you like. It will not be heard. Except by those who are listening. Such simple truths WILL be heard. > > You can repeat your prayer as often as you like. It will not be heard. Except by those who are listening. Such simple truths WILL be heard.
From: Virgil on 13 Nov 2006 15:56
In article <1163428970.972473.215570(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > Every column has a fixed number omega of terms. You can add 1 further > > > term and you will get omega + 1 terms in every column. Every line has > > > less than omega terms. If you add 1 term to every line, the number of > > > terms in every line remains less than omega. What about the diagonal? > > > It has to satisfy both conditions, which is impossible. > > > > No. Adding one element to each line does not change > > the supremum of the lengths of the lines, so it does > > not change the length of the diagonal. > > > > The number of terms in each line, n, is less than omega. > > The number of columns is the supremum of the number of terms > > in each line. The number of columns is omega. > > The number of lines is omega. > > The number of columns is equal to the number of lines. > > > > > > Now add one term to each line. > > > > The number of terms in each line, n+1, is less than omega. > > The number of columns is the supremum of the number of terms > > in each line. The number of columns is omega. > > The number of lines is omega. > > The number of columns is equal to the number of lines. > > PS: You forgot to consider the case that one term is added to each > column. The ordinal of the columns is then larger than omega. But the cardinal is not. Which is what you get. |