Cantor Confusion
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From: Dik T. Winter on 21 Oct 2006 20:37 In article <virgil-9631D5.20420720102006(a)comcast.dca.giganews.com> Virgil <virgil(a)comcast.net> writes: > In article <J7GqFL.9AE(a)cwi.nl>, "Dik T. Winter" <Dik.Winter(a)cwi.nl> > wrote: .... > > You think so. The irrational numbers are defined to be the limits of some > > particular sequences (or rather as equivalence classes of sequences). I > > think you have no idea how numbers (yes, I use that term while you think > > it is disgusting) are defined. I will repeat: > > (1) start with the natural numbers as defined by Peano (you may start with > > 0, 1 or 2, but starting with 0 makes everything a bit easier). > > (2) define arithemetic with those numbers, using the axioms. > > (3) define negative numbers as pairs of two elements, the first is a single > > bit (the sign), the second is a natural number. > > One can extent from the naturals to the integers in other ways as well. > One way is as equivalence classes of pairs of naturals (a,b), a and b > naturals, with (a, b) is equivalent to (c,d) if and only if a+d = b+c. Yes, that is true. Also the order of the introductions can be changed. It is not even necessary to introduce 0 until after you have introduced the positive reals. The basic idea is that it is reasonably easily proven that all those ways to introduce the numbers lead to isomporhic concepts. The only problem about isomorphy is the various ways in which the reals are introduced from the rationals (I have a book explaining four different methods). If I remember right, you need Bolzano-Weierstrass for that. And at some stage ordering needs to be introduced (but that can be early on with the use of the Peano axioms and the concept is the easiest when you do it at that stage). -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 21 Oct 2006 21:24 In article <1161435318.373825.152830(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > If we could not get to omega, we would not need it. > > > > If you do not need it, so be it. Do not use current set-theory and base > > your analysis on whatever is provided with a finitistic view. There is > > *no* reason to attack people for using it, except that it is possibly > > against your ethical views. > > I know that we cannot count to omega! But we use it in definitions of > limits. Yes, and so do set theorists. What is the problem here? > > Yes, that was Cantor's theory, but that was an error, and it was not the > > essence. See my discussion with Dave Seaman about just that point. I > > do not think he still had that position when he wrote his 'Contributions > > to set theory' some 11 years after his 'About infinite linear point-sets', > > page 213 in the 'Gesammelte Abhandlungen'. The essence of his theory was > > that there were sets that could not be put in bijection with the set of > > natural numbers. > > Yes. But this recognition is impossible or useless unless all natural > numbers do exist. Yes, and that is the essence of the axiom of infinity. > Therefore lim [n-->oo] {1,2,3,...,n} = N. Why? Pray define your use of limit in this case. There is no standard mathematical definition of limits of sets. I gave you one possible (I think reasonable) definition, but you rejected it, so now it is up to you to provide such a definition. In mathematics, please. > Therefore lim [n-->oo] {1 + 1/2^2 + 1/3^2 + ... + 1/n^2} = (pi^2)/6. That is because there is a standard definition for such limits. But you state that all naturals do not exist, so you would state that the standard mathematical definition of limits is nonsense? > Therefore in the vase experiment: lim [t-->oo] X(t) = X(omega). Wrong. X(t) has no limit when t grows without bounds, and X(omega) is not defined. X is a function from the (non-negative) integers to the (non-negative) integers, I think. At least, in the way you defined it. But I am not sure. X(t) was 9.t, but you were also talking about 1/X(t). > > > According to Cantor oo denotes potential infinity, omega denotes actual > > > infinity. If all natural numbers exist, then the infinite oo of > > > analysis in our vase problem is exactly the omega of set theory. > > > > No. The oo in standard analysis is still potential. That is, it can not > > be attained. So the limit lim{n = 1 --> oo} 1/n = 0, does indicate a > > potential infinity, not an actual infinity. > > For which natural number n is 1/n = 0? For none. > Potential means always finite. Yes. > lim{n = 1 --> oo} 1/n = 0 proclaims "omega reached". It proclaims nothing of the sort. You want to read more in notation than is present. You should know the argument why that limit is 0, at least I hope so, because it is part of any mathematics curriculum. And you should also know that in that argument only finite numbers are used. > > But you are dishonest in transforming the vase problem (where the answer > > was asked at t = 0) to another problem (where the answer was asked at > > t = oo). > > 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. After the transformation you ask for f(oo), which is *not* legitimate. So the transformation does not simplify notation, it simply transforms the problem to an illegitimate problem. > > Some confusion, I think (and I am myself also guilty). A constructible > > number is a number that can be represented by a finite number of > > additions, subtractions, multiplications and finite square root > > extractions. So you can not prove that a list of constructable numbers > > gives as the diagonal a constructable number. The very reason is that > > taking the diagonal is not a construction according to the definition. > > So is it not. Good heavens, that is unimportant for the present > argument. Every constructed number is an element of a countable set. > The set of all constructed numbers is countable. Every diagonal number > belongs to this set. Pray define what you understand under constructable number. In mathematics there is a precise definition. Do you think that e is constructible? It can easily be constructed using continued fractions. > > On the other hand, I think you are meaning computable > > No, I was not meaning that. I mean: constructed (by list or by > formula). Pray give a precise definition. Not handwaving. What operations are allowed in the construction of a constructable number in you view? As I read this, (pi^2)/6 is a constructable number (in your opinion), because it is constructed with a formula. That deviates from the common mathematical meaning of constructable. Also your formulation makes 0.554445444444444444444445... (which is Liouville's constant with 40/9 added to it) constructible. Which "real" number is not constructable in your view? I still think you mean computable (in the mathematical sense). -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 21 Oct 2006 21:45 In article <1161435575.019298.164830(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1161378187.155995.290420(a)f16g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > It *is* continuous like a staircase. > > > X(t) = 9 for t = 1 until t = 2 where X(t) switches to 18. That is > > > enough to excflude X(omega) = 0. > > > > 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. But do you not see that you are putting the cart before the horse? You state (1) that lim{t -> oo} X(t) != 0 because X(omega) != 0 (2) next you define X(omega) = lim{t -> oo} X(t) Even when we disregard the non-existence of that limit, this is just a self-fulfilling prophesy. > > > > That is still not a mathematical formulation. More is required. You > > > > need to actually state what you mean with 'number of balls in the vase > > > > at noon' or 'natural numbers in the set at noon'. > > > > > > The number of transactions t is then t = omega at noon. > > > > Is *that* a mathematical definition? Pray provide a real mathematical > > definition. > > lim {t --> omega} t = omega Oh. Provide a mathematical definition of that limit, please. In standard mathematics that limit is undefined. > > 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. > > (7) assume sequences of rationals. Create equivalence classes amongst > > those sequences (a_n ~ b_n if |a_n - b_n| goes to 0; but this is > > losely speaking and quite a few other methods are known, all > > equivalent). .... > > So, at what stage in this process is the limit of a function used to > > define the irrationals? > > At (7). The equivalence classes of sequences of rationals with same > limit. Wrong. At that point you can not talk about sequences of rationals with the same limit, because many of such sequences do not have a limit in the rationals. So (7) is formulated as I wrote it (in one of the forms to define the reals from the rationals). It is not the *limit* that is the irrational, it is the equivalence class of sequences. And (again, losely speaking) the equivalence classes are built in such a way that all members of the classe *ought* to have the same limit in the extended system. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 21 Oct 2006 21:53 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 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. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: mueckenh on 22 Oct 2006 03:45
MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > MoeBlee schrieb: > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > I've never seen "potential infinity" or "actual infinity" in any > > > > > textbook I've used. > > > > > > > > So you read not the right books or too few. > > > > > > 'potentially infinite' and 'actually infinite' are mentioned often in > > > philosophy and history of mathematics. But would you please just refer > > > to a single textbook of set theory, analysis, or calculus that gives > > > mathematical definitions of 'potentially infinite' and 'actually > > > infinite'? > > > > In > > fact there are few modern books mentioning the difference. One is > > Fraenkel, Abraham A., Levy, Azriel: "Abstract Set Theory" (1976) > > p. 6 "the statement lim 1/n = 0 asserts nothing about infinity (as the > > ominous sign oo seems to suggest) but is just an abbreviation for the > > sentence: 1/n can be made to approach zero as closely as desired by > > sufficiently increasing the integer n. In contrast herewith the set of > > all integers is infinite (infinitely comprehensive) in a sense which is > > "actual" (proper) and not "potential". (It would, however, be a > > fundamental mistake to deem this set infinite because the integers 1, > > 2, 3, ..., n, ... increase infinitely, or better, indefinitely.)" > > > > and later: "Thus the conquest of actual infinity may be considered an > > expansion of our scientific horizon no less revolutionary than the > > Copernican system or than the theory of relativity, or even of quantum > > and nuclear physics." > > Yes, just as I said, the discussion is about the philosophy of > mathematics and set theory (and, I should add, about informal concerns > and motivations), but there is not, WITHIN the set theory discussed > there, a definition of 'actually infinite' and 'potentially infinite'. Does the study of formal languages really make incapable of understanding plain text? What is written above means: "INFINITY" IN SET THEORY IS ALWAYS "ACTUAL INFINITY". This could be translated as completed or finished infinity but usually is not, because that would deter new students from studying this matter. And most of them never get a grasp of that fact. Regards, WM |