Cantor Confusion
in [Math]
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From: Virgil on 8 May 2007 13:51 In article <1178628964.841396.155730(a)q75g2000hsh.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 7 Mai, 22:59, Virgil <vir...(a)comcast.net> wrote: > > In article <1178565971.031862.318...(a)y5g2000hsa.googlegroups.com>, > > > > > > > > > > > > WM <mueck...(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. > > Oh yes, it is the same arguing. You hope "the infinite" may fulfill > your requirements. But it won't because it can't. It does so very nicely in ZF and NBG. When WM chooses to impose restrictions on ZF or NBG which are not part of them, then his crippled system produces a crippled set theory.
From: Virgil on 8 May 2007 14:32 In article <1178636838.282834.114390(a)w5g2000hsg.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > 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? A formal definition always REPLACES any informal ones, and governs the meaning and usage of the thing defined. And the formal definition requires a function to be a set of ordered pairs (a relation), so that it is first of all a SET, which is quite different from being merely a rule.
From: Virgil on 8 May 2007 14:03 In article <1178629882.324337.100150(a)u30g2000hsc.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 8 Mai, 00:30, MoeBlee <jazzm...(a)hotmail.com> wrote: > > On May 7, 2:42 pm, James Burns <burns...(a)osu.edu> wrote: > > > > > Too late. I'm afraid I've already rushed to look it up. > > > I only found the 1999 edition of Hrbacek and Jech, but perhaps > > > that will be a close approximation of the wording in the > > > 1984 edition. I searched Google Books with > > > Hrbacek Jech set theory function > > > and got the definition of function in the 1999 edition. > > > (Unfortunately, I can't copy and paste from there. Pardon > > > the typos, please.) > > > > > Oddly enough, the definition of function there looks essentially > > > identical to every other (mathematical) definition of function I > > > have ever seen. > > > > > Jim Burns > > > > Thank you for this. > > > > My remarks in this post are upon the supposition that, between the > > older and more recent editions, there is not a material difference on > > this matter of functions.. > > > > >http://books.google.com/books?id=Er1r0n7VoSEC&pg=PA23&ots=1afi1e6mts&... > > > /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/. > > > > Even in that informal description, the authors do not mention a > > "formula" as did WM. > > A procedure or rule is a frmula. > > > > > 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/. > > > > Right, a function is a relation. A function is not itself, as WM > > claimed the authors to represent, a formula, domain, and range. > > 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. > > > > And, moving past the informal descriptions and motivation, here is the > > explicit definition: > > which does not contradict the former. It replaces the informal description, in the sense that whatever satisfies the formal definition but not the casual descriptions is a function wheras whatever satisfies only the casual and not the formal is not a function. That is how mathematics works: formal definitions are the only ones that count in mathematics.
From: Virgil on 8 May 2007 14:00 In article <1178628554.931048.21720(a)y80g2000hsf.googlegroups.com>, WM <mueckenh(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. > > > > > (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. > > So these people are simply wrong. No reason to change the definition > of a function. In that text, at least in a later edition, there is a description and a more formal definition. The definition is that of set theory, and not the one WM claims. > > > > (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. > > Perhaps that's the reason why these proofs are not worth the paper > they are written on? And perhaps, as a non-mathematician, one might even say an anti-mathematician, WM is incompetent to judge the value of mathematical definitions to mathematics. > > > > (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. > > Perhaps that's the reason why you are wrong? And perhaps, as a non-mathematician, one might even say an anti-mathematician, WM is incompetent to assess the correctness of a mathematical definition in a mathematical context. > > > (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. > > Perhaps that's the reason why your proofs are wrong? And perhaps, as a non-mathematician, one might even say an anti-mathemaician, WM is incompetent to assess the correctness of a mathematical proof in a mathematical context. > > > 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. > > Which university did you attend? I attended Harvard and Oxford, among others. > > > > 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. > > That is wrong. Not in mathematics. > > > 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. > > Of course, the formula connects the elements of domain and range to > get a set of ordered pairs. The definition, even in the book WM cited (at least in a later edition) gives the set theoretic definition as its formal definition, and what it uses in all its proofs. > > > > 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. You need it taught by anyone at all with any more mathematical competence than you have yourself. Only the ignorant laugh at their own ignorance. > > > > 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 in error. Actually, The definition is not what you claim it is: > >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 * > > : > > : > > : * 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/.
From: Virgil on 8 May 2007 14:14
In article <1178636669.541556.185570(a)q75g2000hsh.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 7 Mai, 23:42, James Burns <burns...(a)osu.edu> wrote: > > > Too late. I'm afraid I've already rushed to look it up. > > I only found the 1999 edition of Hrbacek and Jech, but perhaps > > that will be a close approximation of the wording in the > > 1984 edition. I searched Google Books with > > Hrbacek Jech set theory function > > and got the definition of function in the 1999 edition. > > (Unfortunately, I can't copy and paste from there. Pardon > > the typos, please.) > > > > Oddly enough, the definition of function there looks essentially > > identical to every other (mathematical) definition of function I > > have ever seen. > > I completely agree. And there is no contradiction between the two > paragraphs they wrote. Except that it is the formal definition, which differs in a number of essentials from yours, that governs. |