Cantor Confusion
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From: G. Frege on 6 Feb 2007 18:03 On Tue, 6 Feb 2007 17:55:46 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >>>>>> >>>>>> Numbers *are* their representations. [WM] >>>>>> >>>>> Numerals are no more numbers than names are people. >>>>> >>>> It seems that this is a common "theme" among cranks: they constantly >>>> mis-take the name with the object denoted by the name. >>>> >>> It is a natural mistake. [...] >>> >> This may very well be the case. But, hell, if you are involved in >> mathematics, logic and/or philosophy _you have to be_ informed, imho! >> >> [The following] really is a fundamental insight: >> "In the sentence the name represents the object. Objects I can only /name/. Signs represent them. I can only speak /of/ them. I cannot /assert them/. [...]" (L. Wittgenstein, Tractatus Logico-Philosophicus, 1921) > > Are cranks "involved"? :) > I mean in some way or other... :-) F. -- E-mail: info<at>simple-line<dot>de
From: Dik T. Winter on 6 Feb 2007 20:52 In article <1170761231.100893.322920(a)v33g2000cwv.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > On 5 Feb., 05:10, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > > What *is* IIII. You never have defined it. You really do not like > > definitions, as they pin down the real meaning. > > IIII is a primitive. Everybody knows it - even without definition by > Peano or Dedekind. That means, we do not need Peano to know small > natural numbers. Perhaps not. But in axiomatic mathematics it is best to define it. Unless you want to do non-axiomatic mathematics, like Cantor. The problem with non-axiomatic mathematics is that proofs are so difficult as nothing is defined and people can talk at cross-purposes. It is quite possible that I have quite different ideas about IIII than you have. And indeed, that is the case. I do not see it as the number 4, but as a sequence of four strokes. In my opinion the successor to that would be four strokes with a fifth stroke starting at the bottom left and ending at the top right. > > > Henry VIII had 7 predecessors --- only including him they were 8 > > > Henries. > > > > You are shifting position again? > > No. I could use IIIIIII but I can use abbreviations like VII or 7. But you do not define your abbreviations. > > When I asked you about what basic > > way, III c IV c V, you answered that I had to continue with IIII, > > IIIII, etc. > > The basic way to establish IV c V is to use the numbers in their basic > form IIII c IIIII. (Numbers *are* their representations.) No. Numbers are abstract entities. Their representations are concretisations of those abstract entities. For a given number there are many different representations possible, and they may even be contradictionary when you mix representations. If you see the greek letter "delta" do you associate that immediately with the number 4? And if you see the letter "labda" do you associate it with the number 30? Nevertheless, they *are* representations. > > > > > Correct, for instance for 1/7. > > > > > > > > And for computable numbers some representation does exist. > > > > > > But this representation does not necessarily enable us to determine > > > the trichotomy relation with numbers which are really numbers. > > > > Perhaps. How do you establish trichotomy between 1/13 and 1/64? > > Are you really going to base-26 to establish that? That would be > > pathetic. > > But it would be possible! It would yield the famous "mathematical > precision" if this could not be establised otherwise. But normally you would use with natural a, b, c and d: a/b < c/d if a * d < b * c as immediately follows from the definition of the rationals (that you *did* omit in your book). > > > > > > > 1) 1 ist eine nat�rliche Zahl. > > > > > > > 2) Jede Zahl a in N hat einen bestimmten Nachfolger a' in N. > > ... > > > > This is a recursive definition of natural numbers. By (1) we have > > > > one natural number, by (2), from that single natural number we get > > > > a lot of other natural numbers. > > > > > > Why do you say N is wrong in (2) but not in (1)? > > > > Where in (1) is N? I do not see N at all. > > "1 ist eine nat�rliche Zahl" means "1 in N". But N is not yet defined in (1). You give meaning beyond what is stated. > The property "being a > natural number" implies the existence of N. How can that be the case if N is not yet mentioned or even defined? > Of course the requirement > to decide whether n is in N is more circular than the statement that 1 > is in N. There is no such requirement at all. As long as you do not define N, you can not use N. Pray read your book were it is done proper. At the end it is stated that "a in N" means "a is a natural number" (but not that "a is a natural number" means "a in N"). > > You stated that when I asked you for a definition. So what is happening > > here? > > You will have read the definitions in my book. "3 is the set of all > sets of 3 elements" is explained in chapter 10. Not yet, I am in chapter 9 now. But that is a ridiculous definition, as that is extremely circular. And with that definition you will not be able to show that 2 is a subset of 3. > > > > You may verify that the rings: > > > > R(N, +, *) and R(K, '+', '*') > > > > are isomorphic. > > > > > > I do know that. But I don't want some isomorphic sets. I want to > > > define *the natural numbers* which are > > > I > > > II > > > III > > > ... > > > > And elsewhere you are using 1, 2, 3, 4, VIII, etc. Pray remain > > consistent. > > There are different forms of expressing natural numbers. I, II, > III, ... is the fundamental one. 1,2,3, ... is a convenient one, in > particular if numbers like 10^1000 are involved. Yes, those are representations. There are umpteen many ways to represent numbers. But the representations are just that, representations. Not numbers. Just as sqrt(2) is a representation of the square root of 2. > > > > Your existence is not a mathematical existence. > > > > > > This form of existence is the only possible existence. > > > > As you do not define your form of existence, > > Read chapter 10 of my book. I still state that that is *not* methametical existence, but I am not yet at that chapter. > > it is impossible to talk about > > it, at least mathematically. > > Present mathematics has nothing in common with existence. You are focussed on your very personal definition of existence. And I think you mean physical existence (without reading your chapter 10 at all). But in that case, the Euclidean straight line has also no physical existance. > > > I gave two definitions: Peano and that with "+1" which is very close > > > to Dedekind's attitude. (I don't know whether he actually created it, > > > but I know that he would have liked it with "+1" as a primitive). But > > > the axioms do not establish any existence, in particular not when you > > > apply Dedekind's definition of what a number is. > > > > Existence is a mathematical thing when you can establish it by axioms > > or through theorems based on axioms. Anything else is merely phylosophy. > > There is something called reality and another thing called matheology. > Both are disjoint. With your reasoning, Euclidean geometry is matheology. -- 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 6 Feb 2007 22:05 In article <1170762893.276254.14320(a)s48g2000cws.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > On 5 Feb., 06:04, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article <1170606249.100767.104...(a)j27g2000cwj.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > > > > On 3 Feb., 04:12, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > > In article <1170413742.648825.136...(a)l53g2000cwa.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > > > > So you say: > > > > > The union of finite trees contains an infinite tree. > > > > > The union of finite chains contains an infinite chain. > > > > > The union of finite paths contains an infinite path. > > > > > > > > Indeed. > > > > > > > How can a union of finite elements contain an infinite element? > > > > Where in the above is there a union of finite elements? > > In all three statements. No. A finite tree, chain and path are all sets of nodes (by your own definition). The elements in the union are *not* the trees, chains or paths, but the nodes. > > Unions are about *sets*, not about elements. > > Every element is a set. There are only sets in ZFC. In some models of ZFC. You can not prove that every element is a set based on ZFC only. > > > The union of finite sets of finite paths > > > {p(0)} U {p(1),q(1)} U {p(2), q(2), r(2),s(2)} U ... > > > is the set of all finite paths > > > {p(0), p(1), q(1), p(2), q(2), r(2),s(2) ... } > > > > Indeed. > > > > > which obviously contains the union of finite paths of the special form > > > p(0), p(1), p(2), .... > > > as a subset. > > > > Wrong. There is no obviously here. > > It can be proven. {p(0), p(1), q(1), p(2), q(2), r(2),s(2) ... } > contains {p(0), p(1), p(2), ... } p(0) U p(1) U p(2) U ... != {p(0), p(1), p(2), ...} As the p(i) are sets of nodes, on the left hand we have a set of nodes, on the right hand we have a set of sets of nodes. The set on the right is a subset, the set on the left is not. A set of nodes can not be a subset of a set of sets of nodes. To abuse notation a bit, the best you could state is: p(0) U p(1) U p(2) U ... = U{p(0), p(1), p(2), ...} where U{a, b, c} is defined as U{U{a, b}, c}, which can be done using the axiom of pairing and the axiom of union. But there is a huge difference between: {a, b, c, ...} and U{a, b, c, ...}. > > The union of sets of elements does > > *not* contain the union of elements. > > The tree contains it. Yes. So what? > > This is abundantly clear from the > > first eight chapters of your book. > > They report the history, not the truth. (The latter you will find in > Chapter 9 and 10). That is your truth. From what I have read until now, chapter 9 contains only handwaving, intuistic reasoning and whatever. But no serious mathematical content, as no definitions are given at all. I am not really surprised. > > > If you say, as you do above, that the union of finite paths contains > > > an infinite path, > > > p(0) U p(1) U p(2) U ... contains the infinite path p(oo) > > > then this is also true for this subset. > > > > Let's get back to the finite. A = {{a, b}, {a, c}, {a, d}} and > > B = {{a, d}, {b, d}, {c, d}}. A U B consists of: > > {{a, b}, {a, c}, {a, d}, {b, d}, {c, d}}. > > It does *not* contain unions of the sets that are elements of A or B. > > Why don't you construct an example which is represented in the tree? Because it is already abundantly clear that your assertion does not even work in the finite case. > {p(0), p(1), q(1), p(2), q(2), r(2),s(2) ... } contains {p(0), p(1), > p(2), ... } As a subset, but not as an element. > > In what way does > > {p(0), p(1), q(1), p(2), q(2), r(2),s(2) ... } > > contain: > > p(0) U p(1) U p(2) U ...? > > In the way the tree contains them as sets of nodes: p(0) U p(1) U > p(2) U ... = {p(0), p(1), p(2), ... } = p(oo) is a subset of the set > of paths contained in the tree. How do you find that {p(0), p(1), p(2), ...} = p(oo)? on the left there is a set of sets of nodes, on the right there is a set of nodes. How can they be equal? I agree that p(0) U p(1) U p(2) U ... = p(oo), (allowing finite paths in p(oo)), but neither is equal to {p(0), p(1), p(2), ...} It is already not true for the finite case, so I wonder why it must be true in the infinite case. > > > If the union p(0), p(1), p(2), ....contains p(oo) then the union > > > p(0), p(1), q(1), p(2), q(2), r(2),s(2), ... > > > cannot miss it. > > > > What *sets* of paths are you using? > > The set of paths contained in the finite tree T(0), the set of paths > contained in the finite tree T(1), ..., the set of paths contained in > the finite tree T(n), and so on. In what way does the union of p(0), p(1), p(2), ..., contain p(oo)? It does not contain it as an element, but it *is* the union. It even does not contain it as a subset, unless you allow improper subsets. > > > Then apply it twice. First you get the single set of all paths, second > > > you get the set of elements. > > > > Which is a set of nodes. > > Correct. And a set of ndes can be a path. And in most cases is not. > > So you are not taking the union but the union of the union. Let us > > analyse. A set of paths is a set of sets of nodes. So let us > > consider the basic two trees (level 1 and level 2). The sets of paths > > are: > > L(0) = {{0}} > > L1: {{0, 1}, {0, 2}} > > L2: {{0, 1, 3}, {0, 1, 4}, {0, 2, 5}, {0, 2, 6}}. > > Two sets. Unite them once: > > {{0, 1}, {0, 2}, {0, 1, 3}, {0, 1, 4}, {0, 2, 5}, {0, 2, 6}}. > > Unite this one, and we get: > > {0, 1, 2, 3, 4, 5, 6}. > > I do not see a path there. > > Every path is a subsets of the union, for example {0, 2, 6} c {0, 1, > 2, 3, 4, 5, 6}. Which path do you miss? None. So every path is a subset of the union of the union of the sets of finite paths. That is *not* what you did state. You stated that every path is an element of the union of the sets of finite paths. > > Yes. Infinite unions can give finite things. But that is not a > > contradiction. > > It shows clearly that not the number of elements is decisive but their > sizes. Prove it. > > You think so, but provide no proof. > > Look at the EIT. > 1 > 11 > 111 > ... > > If every element of the diagonal exists, then every line must exist. > If the diagonal is longer than every natural number then a line must > be longer too. Back at that again. We were not talking about that one, but about another one. Go on with the tree discussion, please. Logic: F. > > > There cannot be an infinite segment {1,2,3,...n} with a last element > > > unless the last element is infinite. > > > > Sure. But again that is not in contradiction to anything I state. > > It contradicts actual infinity. No. But here also you are switching to something different. Pray remain on-topic. > > > The union of the P(i) is the set of all paths. It includes the set of > > > all paths of the form p(n). Why should it not contain p(oo) if the set > > > of all paths of the form p(n) alone contains p(oo)? > > > > I think we are talking at cross purposes. And are misunderstanding each > > other about what the meaning is of the term "the set of all paths p(n)". > > I think P(i) is the set of all paths of the finite tree with i levels. > p(n) is a path like 0.000...0 with n zeros behind the point. The union > of all p(n), i.e., of all nodes 0, is p(oo) = 0.000... So P(oo) is not the union of all sets of finite paths? But P(oo) is the set of the union of all finite paths? There is a difference. > > > And that is a single set containing every element which is in A or in > > > B or in both. > > > The union of some sets of paths is one set of paths. > > > The union of all paths is one set of elements. > > > > Right. So the union of all paths is a set of nodes. But I do not think > > that set is a path. That set is a tree (by your definition). > > That set is a tree but contains the paths as subsets. Yes. But it is not a path. > > > The union of {three paths} and {12 paths} and {17 paths} is a set of > > > paths. These paths are unioned in the tree to give a set of nodes, > > > perhaps including even one or more paths. > > > > This does not make sense at all. > > The union of the two paths in the tree T(1) and the 4 paths in the > tree T(2) and the 8 paths in the tree T(3) is a set of 14 paths (a set > of 14 sets of nodes). The union of these 14 sets is one set of nodes > (the tree T(3)). This set of nodes contains all the 14 paths as > subsets. Yes. So when we take the union of finite sets of paths, and take the union again of the resulting set, we get a set that has the intended paths as subsets. But you claim that the intended path is an element of the union of finite sets of paths. There is quite some difference. The intended path is a subset of the union of the union of finite sets of paths. Not anything more. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Federico Ferreres Solana on 6 Feb 2007 13:44 Call me ignorant, but I don't understand what Cantor is trying to do with the diagonal. FRACT --> NATURALS 0.0 --> 0 0.1 --> 1 0.2 --> 2 0.3 --> 3 ... --> ... 0.9 --> 9 0.01 --> 10 0.11 --> 11 0.12 --> 12 (etc) This uses the same properties than his (or whoever did that) mapping of rationals to naturals, and would cover all fractional numbers. If you don't like the mapping rule (function, operation or whatever)...call it self referential, not an operation...who cares? Each and every fractional has natural associated with it, and no more. Dilbert is happy. If you think not, then show me a number (if it's infinitely large as PI-3, I will of course express it as REVERSE(PI-3), as you don't know what even 0% of PI looks like, except from it's definition). Therefore, "set size" of fractionals = "set size" of naturals, and "set size" of reals = "set size" of naturals SQUARED. That is, similar to R2. Why? Because for each natural part of a real, you have the "set size" of naturals. And why all this mess? Because reals notation have an artifact, the dot or "." in between two numbers. We use them in practice, because a metter can be divided (it is NOT unity) and we NEED cms, but we can still propose the universe is infinetly large. But as we have infinite "precision" (my coloquial name for it having infinite items in the "list") with just naturals, or just fractionals, we do not REALLY need to use any real numbers. Having to use both fractional parts and integer parts, is because we choose the wrong unity (in practice, and theory). If I could define periodic numbers for the naturals, we'd not need the fractionals (we could theoretically get rid of them if we wished), and else, we could just use fractionals, and choose precision correctly. All other differences come from defining naturals and fractionals as "different stuff", when they are not. One is based on the idea that stuff (e.g. segment [0-1] being a pie...) can be divided an infinite number of times with infinite precision (never knowing what the smallest possible share is), and the other, on the idea that there are finitely countable things (else, no number would make sense in practice), but there there could also be an infinitely long distance or weight or "unities". If you start looking as both problems as the same, with a different strategy (fractionals, I can imagine a pie vs naturals, I can imagine "unity" or a single "object" or atom. So yes, there infinities smaller between the reals and the naturals. For example the segment [0-2] or reals is the naturals size * 2. And of course, the even numbers are just "half" of those in the naturals set. There is no magic. If you don't believe this, I don't care. We we die, we'll know. And if we don't, it won't matter at all. Regards, Federico
From: Federico Ferreres Solana on 6 Feb 2007 13:53
Errata: (last line in sequence) ... 0.12 --> 12 sould read ... 0.21 --> 12 FRACT --> NATURALS 0.0 --> 0 0.1 --> 1 0.2 --> 2 0.3 --> 3 ... --> ... 0.9 --> 9 0.01 --> 10 0.11 --> 11 0.21 --> 12 (etc) |