Cantor Confusion
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From: David Marcus on 1 Nov 2006 18:42 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > David Marcus schrieb: > > > > > > > The set of natural numbers is an infinite set that contains only finite > > > > numbers. > > > > > > Please do not assert over and over again this unsubstantiated nonsense > > > (this word means exactly what you think) but give an example, please, > > > of a natural number which does not belong to a finite sequence. If you > > > cannot do so, then it is obviously unnecessary to consider N as an > > > infinite sequence, because all its members belong to finite sequences. > > > > I didn't say anything about sequences, finite or otherwise. So, your > > request is irrelevant to my statement. > > The sequence of natural numbers is not comprehensible in ZFC? Neither > is the sequence of partial sums of a converging series? Nor are the > finite sequences which are called (initial) segments of sequences which > are ordered sets. Also the expression "extended sequence" for an > uncountable ordered set is new to you? Non sequitor. Let's make it simple. I'll give a statement and you say whether you think it is provable in ZFC. Is The set of natural numbers is infinite provable in ZFC? Please answer "Yes" or "No". > > Don't you think that you should label all your posts as > > "NON-STANDARD MATHEMATICS"? > > 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 non-standard meaning on Cantor's > definition of omega as a whole number? Since Cantor predates axiomatic set theory, if you write anything that uses Cantor's definitions without checking whether the definitions are still standard is "Non-Standard Mathematics". If you want to discuss history, that is fine, but you should label your posts as such. This is simple courtesy. If you use words without defining them, readers assume you are using them in their current meanings. If you are using historical meanings, then either say so or use a different word. As for the current definition of omega, Kunen's book is a good reference. According to Kunen, omega is not a natural number. I'm guessing that by "whole number" you mean natural number, but I really don't know, since you seem to have your own language for everything and you never give definitions for any of the words that you use. -- David Marcus
From: David Marcus on 1 Nov 2006 18:58 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > > I know of a really good logician who told me that he is unable to > > > translate my proof into ZFC. > > > > Hardly surprising. If he is truly a good logician, then he didn't want > > to tell you that you were talking gibberish. > > No, he (and he is not the only one to do so) does understand the > arguing but does not know how to translate it into ZF language. That is > all. If your "really good logician" understands the argument, have him post his version of it. > > 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. So, please desist from making such a spurious claim. If you can prove that the real numbers are both countable and uncountable, then that simply shows that the system that you are using for your proofs is inconsistent. It says nothing about the real numbers and countability as defined in ZFC. > > > > OK, let's use our logic (or standard logic or ZFC). The list has > > > > infinitely many lines and columns, so the number of diagonal elements is > > > > infinite. Your statement that the number of diagonal elements is finite > > > > is wrong. You've jumped from "each entry is finite" to "the number of > > > > columns is finite". > > > > > > Please give one letter which requires an actually infinite number omega > > > of columns. > > > If you can't, please stop the nonsense talk about an infinite number of > > > finite numbers. > > > > You seem to have skipped right over my statement that I was using > > standard logic (or ZFC). In standard logic (or ZFC), "the number of > > columns is finite" is not the same as "each entry in the list has a > > finite number of columns". It seems simply bizarre that you would mix > > these up. > > It seems that you do not understand what logic is. That branch which > you call "standard logic" is some kind of theology which is not useful > to promote clear thinking. Whether standard logic/mathematics promotes "clear thinking" is irrelevant to the current discussion. Standard logic is what standard mathematics uses. If you don't want to use it, then say so, but don't persist in claiming that you are making a standard mathematical argument when you are not. It is like being asked a chess puzzle and giving an answer using the rules of checkers or go. > The maximum of a set of finite numbers which has no maximum is simply > not present. It is *not* an infinite number which is larger than any > finite number, because a maximum must belong to the set. If by "maximum" you mean the largest element and by "finite numbers" you mean natural numbers, then it is true (in ZFC and standard mathematics) that an unbounded set of natural numbers does not have a maximum. > And a supremum not belonging to the elements of the set does not yield > a diagonal digit. No clue what this is supposed to mean. Feel free to rephrase using common mathematical terminology and/or define what a "diagonal digit" is. > > > > > If the list consists of finite sequences, then the diaogonal is a > > > > > finite sequence too. Because it cannot be broader than the list > > > > > > > In ZFC or in some other system? > > > > > > Everywhere. > > > > How would you know since you admitted above that your proofs don't work > > in ZFC? Why make claims that you then immediately contradict? > > I know that ZFC claims there are more reals than any countable set has > elements. Hence my proof contradicts ZFC. I'm sure no one here doubts that your proof contradicts ZFC. But, this only shows that you and ZFC are using different axioms or rules of inference. So what? -- David Marcus
From: MoeBlee on 1 Nov 2006 19:22 Lester Zick wrote: > Okay(x). So(x)exactly(x)which(x)are(x)mangled(x)versions(x)of(x) > what(x)you(x)were(x)claiming(x)? I already pointed them out several times. Now, on the one hand you indicate a sense of wanting to move on from the subject, but you also continue to post about it, including asking me a question which would prompt me yet again to post examples of a subject you indicate you want to move on from. You're hopeless. MoeBlee
From: William Hughes on 1 Nov 2006 20:47 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > Correct. And given any set of lines we can find a finite number which > > > is larger. Otherwise, at least one of the lines would contain a number > > > of letters which was larger than any natural number, i.e. which was > > > omega. > > > > But what is important is not the number of letters in any given > > line but the number of lines. These are two different things. > > You say so. I strongly disagree, because the number of every segment of > lines is the same as the maximum of letters in the lines of the > segment. But luckily this dissent is not relevant, because the diagonal > is not only limited by the length but also by the width of a matrix. > > The maximum of a set of finite numbers which has no maximum is simply > not present. > It is *not* an infinite number which is larger than any > finite number, because a maximum must belong to the set. And a supremum > not belonging to the elements of the set does not yield a diagonal > digit. > Each diagonal digit is formed by a line. The number of diagonal digits is the number of lines. The number of lines is a supremum. Clearly the number of lines does not yield a diagonal digit. Let A be the list of all finite strings. Let D be a diagonal formed from this list. The question is "How many digits does D have". It is not enough for you to say that this question does not have a defined answer. You must also show that the usual answer (omega) leads to a contradiction. Let the number of digits in D be X. We don't know if X exists. However, it is clear that if X does exist it must be greater than any natural number. Hence: - X does not exist or - X is at least as great as omega > > > Why do you say incorrect? Given any integer N, we certainly have more > > letters > > than N. > > But the smallest number which is larger than any integer is omega. And > we have excluded a line with omega letters. But excluding a line with omega letters does not mean that D cannot have omega letters. The number of letters in D is less than the number of letters in a line only when this line has more letters than any other line. There is no line with more letters than any other line, so no line we can use to limit D. > > > > > > > > > > Greater than any integer is no integer but only omega (and > > > its successors). > > > > > > > Call the number of letters anything you want. If it exists it must be > > greater > > than any integer. > > Correct. If it exists, it must be infinite (or omega). Only then the > number of lines can be omega. But we have excluded that the number of > letters is omega. > No, we have not excluded omega. > > If it does not exist you cannot use it to limit the > > diagonal. > > If it does not exist, then you cannot build a diagonal of that length. In which case you cannot use the fact that it does exist to show a contradiction. Yes, you can assume the diagonal does not exist. No, you do not get a contradiction if you assume it does exist. > > > > > > So the greatest number of letters the diagonal can have is > > > > greater than any integer. > > > > > > Incorrect. See above. > > > > Your claim is that the number of letters the diagonal can have > > is limited above by the number of letters in the list. > > It is limited by the number of letters in *one line* of the list. > No, it could only be limited by the number of letters in a line that had more letters than any other line. No such line exists. - William Hughes
From: Dik T. Winter on 1 Nov 2006 21:00
In article <sjnfk256m5tau0bmk5u4vjdg53al1f2sc9(a)4ax.com> Lester Zick <dontbother(a)nowhere.net> writes: > On Tue, 31 Oct 2006 02:27:15 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> > wrote: > >In article <md5ak294kscg4uk48a276jktc64lf430rq(a)4ax.com> Lester Zick <dontbother(a)nowhere.net> writes: .... > > > Of course you need a domain of discourse. In mathematics that is > > > established either by true definitions or axioms. But what I was > > > specifically referring to is a process of particular definition rather > > > than general definition. Particular definition is simple enumeration > > > of some quality to be defined as properties of particular objects. > > > >(That may exist or may not exist.) > > Correct except there are two kinds of things which may not exist: > happenstantially non existents and things which cannot exist. I do not know what the distinction would be. > > > A > > > general definition just defines the subject without reference to > > > objects defined in such terms. If I say "infinity is . . ." it's a non > > > specific general definition applicable to whatever domain applies. > > > >But that is not a mathematical definition, but a philosophical definition. > > What makes you think it's not a mathematical definition? Oh, well, please give me such a definition that is mathematical. > And more > importantly what makes you think it's not a true definition whether > mathematical or not? I still do not see the concept of a false definition. If the requirements of a definition are internally inconsistent that means that the definition can not be satisfied. Take again the definition: A z-prime is a Fermat number larger than 65537 that is prime currently it is unknown whether there are z-primes or not, so in your opinion (I think) z-primes may happenstantially not exist. On the other hand, once it has been shown that all Fermat numbers larger than 65537 are composite, the requirements in the definition are internally inconsistent, so at that moment the definition becomes false? > >Without domain of discourse, such a definition makes no sense, > >mathematically. > > Then perhaps mathematics makes no sense mathematically. And why not? > > > Whereas if I try to define the quaity of infinity in specific objects > > > as in "infinity(x) is . . ." I'm stuck with whatever x may turnout to > > > be and without specifying what x is there is no way to determine > > > whether the definition can be correct or not. > > > >This is wrong. You do not define the quality of infinity in specific > >objects, but as a general property within a set of axioms (part of your > >domain of discourse). > > Then why would anyone try to define infinite(x)? That's nonsense. Depends what you are defining. If it is (for instance) a function from the ordinals to the set {false, true}, it would not be nonsense at all. Just another notation for: "infinite: ordinals -> {false, true}" > > As far as I know, "infinity" in mathematics has > >only been defined in the context of topology, where it is the single point > >used in the one-point compactification (but it is long ago that I did all > >this). And from there it has a derived definitions in projective geometry > >(the point at infinity, and also the line at infinity). The term is used, > >losely, in analysis, using limits, but is not really defined there as term. > >You will see oo used in analysis, but when it is used it does *not* mean > >"infinity". When that symbol is used the usage has a specific definition. > > Obviously because no one can define "infinity" in mathematically > exhaustive terms. Within a set of axioms, it *can* be defined in exhaustive terms, without such a set there is no mathematical definition at all. > > > But don't forget the process of comprehension itself improves vastly > > > with time and experience of expression. When I look back over my own > > > posts the level and sophistication of expression and application has > > > improved enormously. > > > >No, there is something else. In time definitions in a particular field > >of mathematics can change over time. So when you are reading old books > >or articles you need to know whether the definitions actually have been > >changed since then or not. > > Sure but that can be a result of improved comprehension of the > subject. That is not the case. The major reason to look at units as not being prime is the simplification of some theorems and definitions. Not because there is more known about the subject. That is always the case when definitions are changed. The old definition was not wrong, but it appeared to become to involve more and more complicated theorems and definitions. So changing the original definition simplifies theorems and other definitions. But consider something else. You have probably learned at school that a (natural) number is prime if and only if it is only divisible by 1 and itself. (And that probably was the original definition.) However, that property (or actual a similar property) has been given a different name, and being prime has been given a new definition. In the naturals the two definitions give the same result, as they form a unique factorisation domain. But in non-UFD's they give different concepts. That change appears at first sight to be merely gratuitous. > > So now, > >1 is a unit and 2 is the smallest prime in the natural numbers. > > And I'm still left wondering why 0!=1. Depends on how you define the ! function. There are a few reasons to define 0! = 1. (1) k! = gamma(k + 1). Euler did show that the gamma gamma(k + 1) = k! for all natural arguments, so it is pretty logical to define: 0! = gamma(1) = 1. And we even get generalisations, like (-1/2)! = sqrt(pi). (2) If we restrict to integers, it is commonly used that sum{k = a ... b} f(k) = 0 if b < a. In the same way it is pretty reasonable to have prod{k = a ... b} f(k) = 1 if b < a. (3) In Taylor series expansions, 0! = 1 simplifies many expressions, like exp(x) = sum{n = 0 -> oo} x^n / n! Ne |