Cantor Confusion
in [Math]
|
From: Dik T. Winter on 8 May 2007 22:42 In article <1178463202.150771.47030(a)y80g2000hsf.googlegroups.com> WM <mueckenh(a)rz.fh-augsburg.de> writes: > On 6 Mai, 04:33, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > > Your conclusion is invalid. Indeed, each node is passed by a > > > > mutlitude of paths. But the path "0.10101010..." is nevertheless > > > > separated from all other paths. > > > > > > No. Only from those few you can ask for. > > > > And those are (in your opinion) only finitely many. Or can I ask for > > infinitely many paths? > > No. You ma yask for a set of infinitely many paths, but not for > infinitely many paths. So in your opinion there are only finitely many paths in the infinite tree. Great. You reject the axiom of infinity, as I wrote already many times before. (Note: reject, not refute.) > > > > So what? For each two different paths it is possible to state a > > > > node where they do separate. For each path there is no node where > > > > it separates from all other paths. There is no contradiction. > > > > > > No. Only from those few you can ask for. > > > > Opinion again. > > It is proven by Cantor that there are uncountably many real numbers. > It is also clear that there are only countably many questions, no? I would think there are only finitely many questions. But all this is not mathematics but philosphy. > > There is a subtle difference. With the diagonal proof we always remain in > > the finite. > > That's why it fails for numbers with infinitely many digits. No. > > With the paths we can also always remain in the finite because > > any two different paths have a node where they differ *in the finite*. So > > for each node there is a path p' that has it in common with p. But also > > for each p' != p there is a node in p that is not in p'. > > No. Only for those p' you can ask for. There must be others, because > there is no node which belongs only to p alone. I do not ask for each individual path. I show it for all. Or do you reject the formula sum{i = 1..n} = (n + 1) * n / 2? You can not ask the validity for all n, but only for finitely many n, according to your thinking. So for each and every n you have to prove it again. > > > "Up to all nodes" means the sequence of nodes has been worked > > > completely. > > > > But infinite paths do not have a final node, so what do you mean? > > I mean that the speech of "all or complete or finished" is in clear > contradiction with "infinite". So your speech "all or complete or finished" (it was yours) is in contradiction with infinite. That is your speech. Adapt your speech if you want to talk mathematics. > > > > > > It is your simplified finitistic view where that > > > > > > means that there is a larger number. > > > > > > > > > > It is the meanig of "completed". > > > > > > > > Your meaning, perhaps. > > > > > > "completed" means nothing remains. > > > > Perhaps. > > Sure. Oh. A mathematical proof, please. > > > In the present state of affairs we know that at every node of path p > > > which we look at, path p is not unique. And we know, that we can test > > > and test for different paths p', but we cannot carry out more than > > > countably many tests (our list is limted). > > > > You can do countably many tests? I thought you could only do finitely > > many tests. > > Please read carefully. We cannot do more than countably many. And in > finite time, we cannot do more than finitely many. Perhaps. You are getting more and more philosophical each turn. > > But again, in mathematics it is *not* necessary to test > > each and every individual. Otherwise you could not even prove that > > sum{i = 1..n} i = (n + 1) * n / 2 > > for all n, but only for finitely many n. > > Of course this is only true for finitely many n. For infinitely many > natural numbers, we get aleph_0. You are wrong. We do not get that. If so, pray show a proof. But your reluctance to show proper proofs and definitions is, by this time, unremarkable. > > But we can mention and test only finitely many paths. So you now say that > > there are only finitely many paths? And that > > sum{i = 1..n} i = (n + 1) * n / 2 > > holds only for those n for which it has been tested? > > It holds for finitely many natural numbers. The natural numbers are a > countable set. Countable != finitely many. So it holds for a finite set. What if we get at a number not in that set; do we need to prove it again? > > > In the present state of affairs we know that at every node we look at, > > > the path p is not unique. > > > > Yes. But to mathematicians that is not a problem. Stronger, try to > > factorise largish numbers without the use of numbers that are not > > representable by a finite number of digits. All those numbers that are > > used in methods like the Number Field Sieve are unique paths within your > > trees, although you are not able to find a distinguishing feature at some > > finite point in your trees. > > Have you ever tried to handle a natural number with 10^100 digits > which are not given by an abbreviation which can be defined by less > than 10^100 bits? Or do you doubt the existence of those natural > numbers? What is the relevance? But try to find the first occurrence where pi(x) and Li(x) switch place. Nevertheless, it *is* a natural number. But I do work with numbers like sqrt(5) without problems. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Michael Press on 8 May 2007 23:37 In article <virgil-D8BB91.20250907052007(a)comcast.dca.giganews.com>, Virgil <virgil(a)comcast.net> wrote: > In article <f1o8hh$5g4$1(a)news2.open-news-network.org>, > Carsten Schultz <carsten(a)codimi.de> wrote: > > > Virgil schrieb: [...] > > > As usual, WM includes only the irrelevant bits and excludes the part > > > that gives the formal definition, and, incidentally, proves him wrong. > > > > So, is he dishonest or just clueless? > > WM has all the earmarks of one of Eric Hofer's "true believers", whom > are, I suppose, closer to clueless than dishonest. What ever it is he chooses to do it, and he knows the meanings of the responses to his acts. Then he lies about these things. Life is a feast and most poor suckers are starving to death. --Auntie Mame -- Michael Press
From: Carsten Schultz on 9 May 2007 00:56 WM schrieb: > On 8 Mai, 17:48, Carsten Schultz <cars...(a)codimi.de> wrote: >> WM schrieb: >> >> >> >> >> >>> On 7 Mai, 23:50, Virgil <vir...(a)comcast.net> wrote: >>>>> http://books.google.com/books?id=Er1r0n7VoSEC&pg=PA23&ots=1afi1e6mts&... >>>>> k++Jech+set+theory+function&sig=Zx5hPqZZ2icNy3mkguHi9kyrFVA#PPA24,M1 >>>>> /Introduction to Set Theory/, by Karel Hrbacek, Thomas J. Jech >>>>> (1999) [pages 23-4] >>>>> : >>>>> : * 3. Functions * >>>>> : >>>>> : 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/. >>>>> : >>>>> : * 3.1 Definition * A binary relation /F/ is called a >>>>> : /function/ (or /mapping/, /correspondence/) if /aFb_1/ >>>>> : and /aFb_2/ imply /b_1 = b_2/ for any /a/, /b_1/, and >>>>> : /b_2/. In other words, a binary relation /F/ is a function >>>>> : if and only if for every /a/ from dom /F/ there is exactly >>>>> : one /b/ such that /aFb/. This unique /b/ is called >>>>> : /the value of F at a/ and is denoted /F(a)/ or /F_a/. >>>>> : [F(a) is not defined if /a [not in] dom F/.] If /F/ is >>>>> : a function with /dom F = A/ and /ran /F/ [subset] B/, >>>>> : it is customary to use the notations /F: A -> B/, >>>>> : /<F(a)| a [in] A>/, /<F_a>_a[in]A/ for the function /F/. >>>>> : The range of the function /F/ can then be denoted >>>>> : /{F(a)| a [in] A}/ or /{F_a}_a[in]A/. >>>>> : >>>> As usual, WM includes only the irrelevant bits >>> I only wanted to avoid typing infinite definitions. >>>> and excludes the part >>>> that gives the formal definition, and, incidentally, proves him wrong >>> The second paragraph proves the first one wrong, in your opinion? >> The second paragraph proves your interpretation of the first paragraph >> wrong. Note that `procedure' and `rule' in the first paragraph are >> undefined, which is why axiomatization of set theory is a good thing. > > Note that formula n my text is as undefined as rule in H&J. > Sure, you like undefined terms, because they allow you to play dishonest semantic games. > Why don't you ask a mathematician how to interpret text, if you can't > read? -- 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: WM on 9 May 2007 06:36 On 8 Mai, 19:01, MoeBlee <jazzm...(a)hotmail.com> wrote: > On May 8, 5:49 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > On 7 Mai, 22:36, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > > (1) Who is the author of that book? I'd like to look it up to see just > > > what is written there. > > > Karel Hrbacek and Thomas Jech: "Introduction to Set Theory" > > Marcel Dekker Inc., New York, 1984, 2nd edition. > > In which book we find the standard set theoretic definition. Your > mistake is in conflating their mention of the general non-rigorous > notion with the actual definition they give. Their mention is nevertheless correct. I quoted them including "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." In order to define that relation, we need a formula or rule or whatever, a domain and a range. > > Those people give the same definition as Hrbacek & Jech, then they give a formula or rule and two sets > you fool. You are loosing self control. Or do you think such expressions will support your wrong claim that H&J would contradict themselves within few lines? Then you are loosing your intellect too. > > > Which university did you attend? > > I did not say that I have or attended or have not attended any > university. It's irrelevent. According to your performance here, in particular your discussion about what a function is and your lack of understanding any textbook, you never saw a university from inside, I suppose. So it is in vain to exchange any further argument with you. Bye. WM
From: Carsten Schultz on 9 May 2007 06:41
WM schrieb: > On 8 Mai, 19:01, MoeBlee <jazzm...(a)hotmail.com> wrote: >> On May 8, 5:49 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: >> >>> On 7 Mai, 22:36, MoeBlee <jazzm...(a)hotmail.com> wrote: >>>> (1) Who is the author of that book? I'd like to look it up to see just >>>> what is written there. >>> Karel Hrbacek and Thomas Jech: "Introduction to Set Theory" >>> Marcel Dekker Inc., New York, 1984, 2nd edition. >> In which book we find the standard set theoretic definition. Your >> mistake is in conflating their mention of the general non-rigorous >> notion with the actual definition they give. > > Their mention is nevertheless correct. I quoted them including "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." > > In order to define that relation, we need a formula or rule or > whatever, a domain and a range. What is the usual definition of a relation? >> Those people give the same definition as Hrbacek & Jech, > > then they give a formula or rule and two sets > >> you fool. > > You are loosing self control. Or do you think such expressions will > support your wrong claim that H&J would contradict themselves within > few lines? Then you are loosing your intellect too. >>> Which university did you attend? >> I did not say that I have or attended or have not attended any >> university. It's irrelevent. > > According to your performance here, in particular your discussion > about what a function is and your lack of understanding any textbook, > you never saw a university from inside, I suppose. > > So it is in vain to exchange any further argument with you. You are in many ways a disgrace to the university from which you graduated. -- Carsten Schultz (2:38, 33:47) http://carsten.codimi.de/ PGP/GPG key on the pgp.net key servers, fingerprint on my home page. |