Cantor Confusion
in [Math]
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From: William Hughes on 7 May 2007 15:49 On May 7, 3:38 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > On 7 Mai, 18:53, Virgil <vir...(a)comcast.net> wrote: > > > > > In article <1178537072.163162.176...(a)u30g2000hsc.googlegroups.com>, > > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > On 7 Mai, 00:59, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > Look! Over there! A pink elephant! > > > > That seems necessary to make us believe that a paths that shares > > > every node with another path does not share every node with another > > > path. But it is not sufficient. > > > > Regards, WM > > > If WM believes that one path in any tree can share every node with > > another and different path, he has passed beyond reason into some sort > > of fugue state. > > Can you explain, how every node of p is shared by other paths, but not > every node is shared by at least one path p'? Yep. An uncountable number of paths share only the first node of p An uncountable number of paths share only the first two nodes of p An uncountable number of paths share only the first three nodes of p ..... Since there is no last path which branches off, this list never ends. So every node of p is shared by other paths. However, every path branches of somewhere, so there is no path p' which shares every node of p. - William Hughes
From: Carsten Schultz on 7 May 2007 16:34 WM schrieb: > On 7 Mai, 18:50, Virgil <vir...(a)comcast.net> wrote: >> In article <1178536779.509069.124...(a)l77g2000hsb.googlegroups.com>, > >>>> That is a very common NON-formal definition and notion of function >>>> found in a great amount of mathematics. However, it is NOT the set >>>> theoretic defintion that is being used in formal Z set theory and is >>>> NOT the definition that is used in formal Z set theory to prove >>>> Cantor's theorem that there is no function from a set onto its power >>>> set. >>> It is a definition from a book on set theory, called "Introduction to >>> Set Theory". So it gives basic set theory. >> It may very well be a very naive set theory for non-mathematicians. >> Who wrote it, and for what sort of students is it supposed to be an >> introduction? > > Karel Hrbacek and Thomas Jech: "Introduction to Set Theory" > Marcel Dekker Inc., New York, 1984, 2nd edition. 250 pages. > For students of set theory. Now please, before we rush to look it up: I guess that everyone here agrees that a function is a special kind of relation, where a relation is a subset of a cartesian product of two sets. What is debated is that there is an additional requirement that a function has to assign values according to a rule in a sense which implies computation, definability or something similar. So what exactly is it that the book you mention says? And when you answer, please distinguish between a technical definition and illustrative prose. Thank you, Carsten -- Carsten Schultz (2:38, 33:47) http://carsten.codimi.de/ PGP/GPG key on the pgp.net key servers, fingerprint on my home page.
From: MoeBlee on 7 May 2007 16:36 On May 7, 4:19 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > It's not insane just to point out that a proof in formal Z set theory > > is one that uses only first order logic applied to the axioms of > > formal Z set theory. > > I only corrected your misprint which you deleted here. Actually, in your reply, YOU deleted the SUBSTANTIVE comments I made in that part of my post, thus you did not respond to that part of my post, but instead rested with the joke of writing 'insane' in reply to inane', which is quite inane. So, again, for the third time, here's what I wrote, which you did not address: "(1) If a presentation of a proof uses any principles or premises whatsoever other than first order logic applied to the Z axioms, then that presentation is not that of a Z set theory proof. So, to reiterate, since you are so very obtuse, if you find that some presentation or another uses some premises or principles other than first order logic applied to the Z axioms, then that presentation is not of a Z set theory proof. (2) Usual textbook presentations of the theorem that no set maps onto its power set use nothing but first order logic applied to the axioms of Z set theory." > > > > (3) In ordinary set theory, a function is a certain kind of set of > > > > ordered pairs and, in ordinary set theory, a function is NOT a triple > > > > of a formula, domain, and range. That you even think a set theoretic > > > > function must have a FORMULA (!) as a component shows, again, > > > > your > > > > lack of understanding even the basics of set theory. > > > > Function, as understood in mathematics, is a procedure, a rule, > > > assigning to any object a from the domain of the function a unique > > > object b, the value of the function at a. A function, therefore, > > > represents a special type of relation, a relation where every object a > > > from the domain is related to precisely one object in the range, > > > namely, to the value of the function at a. > > > That is a very common NON-formal definition and notion of function > > found in a great amount of mathematics. However, it is NOT the set > > theoretic defintion that is being used in formal Z set theory and is > > NOT the definition that is used in formal Z set theory to prove > > Cantor's theorem that there is no function from a set onto its power > > set. > > It is a definition from a book on set theory, called "Introduction to > Set Theory". So it gives basic set theory. It shows that your > statement > "> That you even think a set theoretic > > > function must have a FORMULA (!) as a component shows, again, your > > lack of understanding even the basics of set theory." > > is as wrong as most of your statements. (1) Who is the author of that book? I'd like to look it up to see just what is written there. (2) You can look at the most commonly used and respected textbooks in set theory to see that they don't define a function as a formula, domain, and range. Enderton; Suppes; Levy; Quine; Bernays; Kunen; Stoll; Halmos; Moschovakis; Jech; Takeuti & Zaring; Shoenfield (in the set theory chapter); Chang & Keisler (in the set theory chapter); Mendelson (in the set/class theory chapter), Godel (in his small book) do not define a function as a formula, domain, and range. (3) In such textbook proofs of Cantor's theorem (the theorem that no set maps onto its power set), the definition of 'function; used is the one I've mentioned and is not that of a formula, domain, and range. (4) No matter what ANY author says, the formal Z set theory proof that I point to (which agrees with those authors anyway) is using the definition of 'function' that I mentioned and not that of a formula, domain, and range. (5) Taking a function to be formula, domain, and range in a formal proof of Cantor's theorem would be NONSENSE. We don't do that. We take a function to be just what I've said it is and we prove that, for all x, there is no function on x and onto Px. And if you think that that proof uses the definition of a function as being a formula, domain and range, then you have NO IDEA what the proof is. I said that the proof in formal Z set theory of Cantor's theorem uses nothing but first order logic applied to the axioms of set theory. You have not refuted that, and you could see for yourself that it is true if you only knew basic predicate calculus and even less than what one learns by the end of a first semester course in set theory. > > If you INSIST on using 'function' in a sense that is NOT the sense in > > formal Z set theory, then you are not talking about formal Z set > > theory. > > I never wanted to talk about astrology, why should I talk about formal > set theory? You WERE talking about when you tried to dispute me as to proving Cantor's theorem in formal set theory. And when you're talking about set theory in general, if you're not talking about with an understanding that it has a formal version, then your remarks pertain only to informal versions of set theory that have LONG AGO been supplanted by the rigorous formalization as a formal first order theory. And that is what is at stake in such issues as to whether there are "hidden assumptions" (or whatever phrase you used). > > > Please do not believe that I will repeat for another 1000 postings > > > your controversy with other great mathematicians about the properties > > > of a function. > > > All you need to do is pick up a textbook on set theory to see that in > > ordinary set theory 'function' is defined to be a certain kind of set > > of ordered pairs and not as you define. > > Did I oppose to a function being a set? I don't know why I should have > done that. Perhaps your understanding is not as sharp as you think? By saying that a function is a formula, domain, and range you contradict the set theoretic definition of a function being a certain kind of set of ordered pairs. And, as to understanding, notice that I did NOT say that you contradicted that a function is a set, but rather I am informing you that in SET THEORY (such as ordinary textbook set theory and as formalized as formal first order theory) a function is a certain kind of set of ORDERED PAIRS, and you contradict that as you take a function to be a formula, domain, and range, since whatever that is, either formalized as a triple or not formalized at all, it is not a set of ordered pairs and, a fortiori, not the kind of set of ordered pairs that a function is in set theory. > > No, a function is not ITSELF a triple of a formula, domain, and range, > > but a formula may DEFINE a certain set of ordered pairs that is a > > function. You need to understand such basics of set theory. > > Do I need that, in fact? Best taught by you? Amusing. Apparently you do! You claim that in set theory a function is a formula, domain, and range. So you DO need to be told that in set theory that is NOT what a function is. > > > > > You are wrong to assume that I did not study set theory. > > > > > I infer it by the ignorance you display. > > > > Obviously you are not very good in concluding from given facts on > > > hidden facts. > > > I have no idea what "hidden facts" you're thinking of. > > I see. You see; who else does? You see "hidden facts" but only respond with "I see" when I mention that I have no idea what "hidden facts" you're thinking of. Indeed, you really are a crank. "Hidden facts". Positively risible. > > > > If you want to talk about an infinite summation, then you need to be > > > > able to eliminate the '...' and put it in the form: Sum(from n to oo) > > > > F(n). > > > > Sum [n in N] (2-1-1) < 3. > > > That is but another way of writing 2-1-1+2-1-1+2-1-1+-... < 3. > > > That is not a DEFINITION of "2-1-1+2-1-1+2-1-1+-... " > > Why do you dislike it? It's not a matter of dislike. It's just that it's not a definition! > There are definitions like Sum [n in N] (1) = aleph_0 in several books > on set theory. You can mention anything in particular so that I can evaluate the context of the notation. > Do you miss the sentence: let n be a finite ordinal, and let N be the > set of all finite ordinals, then ...? That doesn't lend a definition to: "2-1-1+2-1-1+2-1-1+-... " > > > If you have problems with infinite sums or products then you should > > > look up set theory, for instance Koenigs theorem. > > > I have no problem with infinte sums and products. I am just pointing > > out that your notation is NOT justified as infinite summation until > > you specify what F is in Sum(from n to oo) Fn. > > Fn = (2-1-1) My notational slip actually. I meant to write: Sum(n=0 to oo) Fn or whatever range you wish instead of starting at 0. Anyway, your Fn = 2-1-1 is just a constant function! What infinite summation do you think you get from a constant function?! Sheesh! > It is the number of separated path coming out of the n-th element of > the tree minus the number of ingoing separated paths minus the number > of nodes of this element. And you just said that for n it is 2-1-1. So it's a constant function. So, for an infinite summation, just do the correct math given F is a constant function! What do you think Sum(n=0 to oo) 2-1-1 = Sum(n=0 to oo) 0 is? > > As I said, set theory is not answerable to your fingerpainting with > > symbols. Nor am I. > > It seems to be a rather sad theory. Since you don't even know its basics, your comments on it are rather pointless. MoeBlee
From: Virgil on 7 May 2007 16:55 In article <1178565586.844949.42730(a)n59g2000hsh.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 7 Mai, 17:46, Virgil <vir...(a)comcast.net> wrote: > > In article <1178532990.891073.253...(a)y5g2000hsa.googlegroups.com>, > > > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > On 6 Mai, 20:16, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > On May 6, 10:53 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > > > Fine. That means, the set is countable but there is no bijection with > > > > > N definable. > > > > > > No it means that while a bijection exists it is not finitely > > > > definable. > > > > > The bijection with this countable set is not finitely definable. > > > Why then do you require a finitely definable bijection between paths > > > and nodes in the tree? > > > > > You are the one pushing "finitely definable", whatever that means. > > That means: You cannot determine or define or construct or write down > or tell me a bijection between the set of numbers which are definied > by a definition identifying them uniquely and the set of natural > numbers. In Zf, every set, including N, has a power set, and there is provably no surjection from any set in ZF to its power set, but trivially an injection, so that the cardinality of any set is less that that of its power set. The paths in a complete infinite binary tree in ZF are easily seen to biject with the power set of N, so that set of paths has cardinality greater than N. And nothing that WM can say can change that. > > > > But as the construction of the Cantor diagonal is finitely defineable, > > and finitely defined, there should be no objection to it. > > That is not the point here. The point is that there are subsets of > countable sets which do not allow for a bijection with N. Translate > this fact to the binary tree. Finite subsets of countable sets are the only subsets of those countable sets which "do not allow for" a bijection with N, at least in ZF. I terms of a binary tree each of the uncountably a many subsets of N determines a unique path wish branches left from each level indexed in that set of naturals and right at each level no in that set. That achieves precisely the requested translation.
From: Virgil on 7 May 2007 16:59
In article <1178565971.031862.318030(a)y5g2000hsa.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 7 Mai, 18:06, Virgil <vir...(a)comcast.net> wrote: > > In article <1178533278.892846.274...(a)w5g2000hsg.googlegroups.com>, > > > > > > The number of separated paths is less than the number of nodes at each > > > finite level. What happen after each finite level is irrelevant. > > > > The number of paths, or more properly uncountable sets of paths, > > distinguishable at any level is, like the level itself, finite, but > > there are infinitely many levels, and what happens at any one level is > > not the end. In order to consider the whole tree one must consider all > > infinitely many levels. > > Yes. But if you consider two infinite sequences, namely the sequence > 1,2,3... and the sequence 2,4,6,..., and put the tems in bijection, > 1-2, 2-4, 3-6, ..., then you can wait and wait and wait: The first > term will never get larger than the second. Even in the infinite this > will not happen. What you are saying has no relevance to paths in my infinite trees. As paths are determined by whether they branch left or right at a given level, one only needs to know the set of levels at which a path branches left to know the entire path. > > It is a pity. It is certainly a pity that WM doesn't understand this stuff. |