Cantor Confusion
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From: David Marcus on 23 Oct 2006 23:23 MoeBlee wrote: > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > Okay. Define your "identity relation". > > If it's primitive, then it's not defined. Usually, in mathematical > logic, the symbol for identity is the only primitive predicate symbol. > And it cannot be defined, even if we wanted to, in first order logic, > which goes along with what I mentioned in my earlier post. > > Again, I have to say, it is not efficient for me to answer questions > going in reverse. That is, if you say, "Why?" or "Define", then you can > ask "Why?" of "Define" after every of my answers to a previous question > of "Why?" or "Define". And I can keep answering until I finally arrive > at the primitives of the theory or even the primitive notions of the > informal meta-meta-theory. But it is much more efficient for us to > START with those primitives and then demonstrate theorem by theorem. > > Thus, I suggest you just get a book on the subject and learn it like > anyone else has ever had to learn a technical subject. Then, I hope > that I could help you with any questions you have along the way. Indeed. One cannot learn mathematics by arguing with people on sci.math. -- David Marcus
From: Randy Poe on 24 Oct 2006 00:22 Lester Zick wrote: > On 23 Oct 2006 08:48:07 -0700, "Randy Poe" <poespam-trap(a)yahoo.com> > wrote: > > > > >Lester Zick wrote: > >> On Mon, 23 Oct 2006 02:00:06 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> > >> wrote: > >> > >> >In article <1161518242.756958.103660(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > >> > >> [. . .] > >> > >> > Virgils definition, although not wrong, definitions > >> >can not be wrong, does not lead to desirable results. > >> > >> Since when can definitions not be wrong? > > > >A definition is a statement by a person that a certain > >symbol will be used by them to stand for some concept. > > > >What is there in such a statement that can be wrong? > > The concept? > > >Example: I will use the term "gleeb" to refer to an integer > >which is divisible by 2. > > > >How can that statement be wrong? How would you > >define "wrong" for such a statement? > > Self contradictory predicates defining the concept in the definition. > Ex: "squircles are square circles" "x is an even, odd" "gleeb is a > finite integer divisible by 0". All of those are perfectly valid definitions. Just because it doesn't exist doesn't mean the concept can't have a name. - Randy
From: mueckenh on 24 Oct 2006 05:50 Dik T. Winter schrieb: > In article <1161517636.934369.301190(a)i42g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > > > > lim{n --> oo} 1/n = 0 proclaims "omega reached". > > > > > > It proclaims nothing of the sort. You want to read more in notation than > > > is present. > > > > You want to read less than present and necessary. > > lim{k --> oo} a_k = pi means omega reached. Or would you state that pi > > belongs to Q like every a_k for natural k? > > Of course not, and of course I do not state that oo is reached. Stating > (for the first limit above) that omega is reached is tantamount to stating > that there is an n such that 1/n = 0. Your logic is strange. If there were an n with 1/n = 0, then we would not need to write "limit". > There *is* no such n. Therefore we write the limit. The limit of all n is omega. > second statement it would mean that there is a k such that a_k = pi. > Both are nonsense. Because there is not such an n, we use the limit. > > > > > It is nothing but just a simplification in notation! > > > > > > It is not. When giving a function from R to R, f(0) might exist, but > > > f(oo) *never* exists. So asking for f(0) in the first case is a > > > legitimate question. > > > > Not more legitimate than asking for f(oo) because at noon also > > infinitely many transactions must have happened. Therefore the > > distinction you make is false. > > The distinction I make is *not* based on the number of transactions. But the *facts* are based on the number of transactions. The trick with the vase is simply to force the infinite number of transaction into a finite time interval. > It is > > The problem is *not* with the number of transactions. You are wrong, but I don't like to continue this idle discussion. > > > With straight-edge and compass. Or else, please solve the problems of > 'doubling the cube', 'squaring the circle' and 'trisecting an angle', > using straight-edge and compass. Apparently you are back to 15-th > century mathematics. It is ridiculous to see how you willful try to misunderstand me. > > > Of couse 0.110001000... and all its sums with rational numbers are > > constructible numbers in my sense. > > But the constructable numbers in your sense are not countable. They are, because there can be not more than countably many constructions. König and Cantor did not yet know the constructible numbers in modern sense. Nevertheless they were convinced that only such numbers which can be constructed in my sense are meaningful. The reason is the countable set of finite constructions. A real number with no definable law cannot be constructed because an infinite amount of information would be needed. Every construction is finitely defined. König used this argument to show a fault in set theory: Man zeigt sehr leicht, daß die endlich definierten Elemente des Kontinuums eine Teilmenge des Kontinuums von der Mächtigkeit aleph_0 bestimmen, ... J. König: Math. Ann. 61 (1905) 156 - 160 Über die Grundlagen der Mengenlehre und des Kontinuumproblems. Cantor recognized the truth of Königs conclusion but doubted that the finitely defined numbers were countable: Wäre Königs Satz, daß alle "endlich definierbaren" reellen Zahlen einen Inbegriff von der Mächtigkeit aleph_0 ausmachen, richtig, so hieße dies, das ganze Zahlenkontinuum sei abzählbar, was doch sicherlich falsch ist. Cantor to Hilbert. I think today there is no question that the constructed numbers (in my sense) are countable. And of course there is no question that König was right. Regards. WM Regards, WM
From: mueckenh on 24 Oct 2006 05:57 Dik T. Winter schrieb: > In article <1161518008.776999.238550(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > In article <1161435575.019298.164830(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > ... > > > > > How do you *define* X(omega). As far as I know X is only defined > > > > > for real numbers, and omega is not one of them. And I see no reason > > > > > to exclude X(omega) = 0, = 1, = -1 at all from this reasoning. > > > > > > > > lim {t --> oo} X(t) = X(omega) > > > > > > Yes, if you define X(omega) like that your conclusion is obvious. > > > > Not I define omega as the limit ordinal number. That is a matter of set > > theory. > > You use oo, which is *not* the same as omega. oo is not the same as omega. Correct. But if omega exists, then the limit n --> oo is omega. > But when we ignore that > difference, more problems prop up. X(t) was a function from R to R, I > think, not a function from ordinals to something else. But you stated > that X(t) was 9.t. Again lacking precision. Absolute precision. X(t) = 10*t would be a bit sloppy. > On the one hand, > if X(t) = 9.t, we get: > lim{t -> omega} X(t) = X(omega) > but with X(t) = t.9, we get: > lim{t -> omega} X(t) != X(omega). > Yes, in the ordinals multiplication is not commutative... lim {n-->oo} n = lim {n-->oo} 9n = omega. > > > Not I define omega as the limit ordinal number. That is a matter of set > > theory. I only use the continuity of a function which is already > > required to find lim 1/n = 0. > > There is no continuity required to find such a limit. Limits are independent > on continuity. The function: > f(x) = 1 when x = 0 and = 0 when x != 0 > is certainly not continuous at x = 0, nevertheless the limit for x -> 0 > exists, and is 0. What continuity do you use when you calculate: > lim{x -> 0} f(x) > ? If f(0) is undefined, i.e., if 0 is not element of the domain of f, then we can find f(0) = 0 by l'Hospital' s rule. Ever heard of? In your example we have no limit because of lacking continuity. For every interval delta covering the value x = 0, we cannot find an epsilon < 1/3. And as you may know, there is no x > 0 which is closest to 0. > > > > > lim {t --> omega} t = omega > > > > > > Oh. Provide a mathematical definition of that limit, please. In standard > > > mathematics that limit is undefined. > > > > Cantor used omega with two slightly different meanings. omega is the > > set N and omega is the first infinite ordinal number, i.e., the > > smallest number larger than any natural. These two definitions yield: > > lim {t --> oo} t = omega > > and > > lim {t --> oo} {1,2,3,...,t} = omega. > > Yes, so what? I asked for a mathematical definition, not for handwaving. > And none of the usages of Cantor do in any way define the limit. What > *is* your mathematical definition of that limit? Learn mathematics, then you will understand the mathematical definition of the set of all natural numbers as imagined by Cantor and unchanged until toda: lim {t --> oo} {1,2,3,...,t} = omega. > > > > > You think so. The irrational numbers are defined to be the limits > > > > > of some particular sequences (or rather as equivalence classes of > > > > > sequences). I > > > > > > > > Equivalence classes of sequences with same limit like > > > > lim {t --> oo} a_t. > > > > > > Wrong. > > > > Wrong is wrong. The limit *is* the irrational number. You can use > > these and only these numbers in a Cantor list, not the equivalence > > classes of sequences. > > You really do not understand how the reals are defined. The limit is > *not* the irrational number. The limit does not even exist. Therefore you cannot use the reals in a Cantor list. They even don't exist. > > > Pray re-read what I wrote. Why? You do not explain how the reals are used in a Cantor list. > The real numbers are defined as equivalence > classes of sequences of rational numbers. The sequences do not belong to > Q. They are sequences of elements of Q. So when defining reals (according > to this methodology) we start with sequences of rationals. We call two > sequences equivalent if their difference goes to 0, the concept of limit > has not yet even been defined. Without Cauchy convergence you cannot prove equivalence. > It is easily shown that that is an > equivalence relation, so we can divide the sequences in equivalence classes. > Each of those equivalence classes is a real number. So a real number is > an equivalence class of rationals. Equivalence classes cannot be defined without Cauchy convergence, which is an improved definition with respect to the simple limit. Equivalence classes cannot be used in a Cantor list, because the terms of he sequences differ from one another. > It is *extremely* losely speaking. The real (not irrational) number is > an equivalence class, it is not a limit. So initially, 1/2 is *not* > an element of this new system. An element of this new system is an > equivalence class of which the sequence: > 1/2, 1/2, 1/2, ... > is a representative. What about the Cantor's list in this case? > > > You > > can use this and only this number in a Cantor list, not the equivalence > > class of sequences (because the due terms are not uniquely defined). I see. But you should try to understand. In Cantor's list, there are limits of sequences, not equivealence classes. > Oh, perhaps, I do not understand at all. I do not see the relation. > > > Summarizing the original question: You need the limit omega to > > construct the irrational numbers. > > Where in the construction above did I use the limit omega? You will need it in order to construct a real number and its decimal representation for a Cantor list. Regards, WM
From: mueckenh on 24 Oct 2006 06:01
Dik T. Winter schrieb: > In article <1161518242.756958.103660(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > In article <virgil-E8EF11.13483421102006(a)comcast.dca.giganews.com> Virgil <virgil(a)comcast.net> writes: > > > > In article <1161435318.373825.152830(a)e3g2000cwe.googlegroups.com>, > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > ... > > > > > Therefore lim [n-->oo] {1,2,3,...,n} = N. > > > > > > > > Depends on how one defines lim [n-->oo] {1,2,3,...,n}. It is certainly > > > > true if one takes the limit to be the union of all of them, as > > > > guaranteed by the axiom of union in ZF. > > > > > > I would not like that as a definition. That would make: > > > lim{n -> oo} {n, n + 1, n + 2, ...} = N > > > > Obviously wrong, because 1, 2, 3, .., n-1 all belong to N > > Eh? Using the definition Virgil gave leads to my conclusion without further > ado. And I just stated that it was not desirable. Obviously wrong is > something else. If Virgils definition is the definition, my conclusion > is obviously right. So there should be a search for a definition that > allows what is wanted. Virgils definition, although not wrong, definitions > can not be wrong, False. But idle dscussion. >does not lead to desirable results. > > > lim [n-->oo] {-1,0,1,2,3,...,n} = N > > > > is obviuously wrong too. > > Depends on how you define the limit. > > > Therefore lim [n-->oo] {1,2,3,...,n} = N. > > Depends on how you define the limit. That *is* the definition of the this limit. It is a wrong definition only in case N does not actually exist. > > > > I gave sometime ago a definition of the limit of sets that is (in my > > > opinion) workable, but Mueckenheim did not allow that definition. The > > > reason being that under some formulations of the vase problem that > > > definition would make the vase empty at noon. > > > > So it must be wrong and needs no further attention. Why? > > But you never give a definition of the limit of sets. You only state that > definitions are *wrong* (although I fail to see why a definition can be > wrong). But you never state a proper definition. Look here: lim [n-->oo] {1,2,3,...,n} := N. > > > Because the contents of the vase increases on and on. Such a process > > cannot lead to emptiness in any consistent system - independent of any > > "intuition". > > That requires proof. LOL. Idle discussion. Regards, WM |