Cantor Confusion
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From: Dave Seaman on 1 Feb 2007 18:53 On Thu, 01 Feb 2007 21:52:26 GMT, Andy Smith wrote: > Dave Seaman <dseaman(a)no.such.host> writes > (snip) >>> I had previously imagined that the "Continuum Hypothesis" related to >>> whether the set of reals (as 0 dimensional points, each with no >>> neighbouring point) could cover the line, with dimension 1 (Which seemed >>> like a good question). >>You are asking whether R and "the real line" are the same set. In order >>for that question to have a nontrivial answer, you must have some meaning >>in mind for "the real line" other than R. What is your definition? > Well until recently my naive view was that points and lines were > different entities, and you might as well ask how many metres in an acre > as how many points in a line. I would have said (in your terminology) > that there is clearly an injection from the reals into the line, but > that you can never cover it. But I doubt now if that is true, or if the > question is a sensible one to start with. I don't follow. Is it your claim that: (1) A line does not have any points as members at all, or (2) A line has points as members, but perhaps there aren't enough points to fill it? Your metres-in-an-acre analogy seems to suggest (1), but otherwise you seem to be saying (2). If (2) is your claim, then can you show me a point on the line that is not represented by a real number? If you can do this, then evidently "the line" to you must be something other than simply "the set of real numbers", as it is to most people. Can you elaborate on this? -- Dave Seaman U.S. Court of Appeals to review three issues concerning case of Mumia Abu-Jamal. <http://www.mumia2000.org/>
From: Dik T. Winter on 1 Feb 2007 20:19 In article <1170348341.624257.130820(a)p10g2000cwp.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > On 1 Feb., 14:15, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article <1170330236.064219.62...(a)a75g2000cwd.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > > > If you have a set and you add something to this set (in unary > > > representation), then you have a super set of the former. The former > > > is a subset of the latter. > > > > This is not defining what I asked for. What are the elements of the sets > > you are talking about? > > The type of element is not restricted other than by the requirement > that it be part of the real world. We were specifically talking about the sets I, II, III, etc. Or about 4 and 5. I ask you how you define them as sets. But let me elaborate more on this below: > > > > > > Bizarre. So 3 is the set of all existing (what does that mean) > > > > > > sets with 3 elements, If 4 is the set of all existing (whatever that does mean) sets with 4 elements and 5 is the set of all existing sets with 5 elements, we find immediately that 4 is *not* a subset of 5. With this definition of numbers as sets subsetting does not give what you wish. You should (at the least) be consistent. > > > It is circular. But please do not forget that I do not *define* a > > > natural number! I leave that to Peano etc. I look for its *existence*. > > > > So it was not a definition at all. Note also that the Peano axioms to > > *not* talk about representation. > > Therefore we have to investigate whether such a representation exists. > Note that I use the definition of a number by Peano in order to look > for the existence of that number. What does the latter sentence *mean*? But indeed, for most numbers some representations do not exist, while other representations do exist. The existence of a number is independent of the existence of a representation. For instance, for most rational numbers a decimal representation does not exist. > > > Of course it is. You must start at some point and eventually you will > > > come back. Circular reasoning cannot be avoided. But that is not a > > > problem when existence is concerned. The definition of a number is > > > given on p. 3 and p 130 of my book. > > > > A definition that uses circular reasoning is not a definition. If a write: > > a foo is a foo > > The definition of a number is given as non-circular as possible: p. 3: > 1) 1 ist eine nat�rliche Zahl. > 2) Jede Zahl a in N hat einen bestimmten Nachfolger a' in N. in N is wrong here; N is not defined. A more proper version is: 2) Jede nat�rliche Zahl a had einen bestimmten Nachfolger a', und dass ist auch ein nat�rliche Zahl. > 3) Es gibt keine Zahl mit dem Nachfolger 1. > 4) Aus a' = b' folgt a = b. > 5) Jede Menge M von nat�rlichen Zahlen, welche die Zahl 1 und zu jeder > Zahl a in M auch den Nachfolger a' enth�lt, enth�lt alle nat�rlichen > Zahlen. Yes, I know Peano pretty well. But this is *not* a circular definition. I wonder why you think it is circular. > p. 130: > > 1) 1 in M. > 2) If n in M then n + 1 in M. I prefer the successor of n, rather than n + 1 here. > 3) IN ist Durchschnitt aller Mengen M, die (1) und (2) erf�llen. That is an alternative definition, again not circular. > To investigate the *existence* of number 3 we can use the definition > of 3. And *in this context* the definition of 3 is: succ(succ(1)) no set at all. And the existence of 3 follows from the first set of axioms you gave. Your definition is circular, this one is not. -- 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 1 Feb 2007 20:42 In article <1170348637.768496.237360(a)a75g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > On 1 Feb., 14:09, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article <1170328870.612961.224...(a)v33g2000cwv.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > > Why do you accept then that the union of sets of nodes (finite trees) > > > establishes an infinite set of nodes (an infinite tree) without such > > > an infinite set (infinite tree) being contained in the union? > > > > Because that is something different. The union of sets of paths > > establishes an infinite set of paths. But that infinite set of > > paths does not contain an infinite path, because none of the > > constituent sets contains one. > > So the union of finite chains of nodes like the following > 0 > 12 > 6543 > does not contain an infinite chain of nodes of the form > 0 > 12 > 6543 > 789... > ??? I did not state that. Pray try to read what I wrote, rather than giving your own interpretation to it. > Frankly, I do not see much difference to: The union of finite paths > contains an infinite path. But that is true. What I state again, again and again, but what you misread each and every time is: "the union of sets of finite paths does not contain an infinite path". > > > How then does this infinite path manage to enter the union of the > > > finite trees? > > > > Because the set of paths of the union of finite trees is *not* contained in > > the union of the sets of finite paths. > > And it is not contained in any of the finite trees. Why is it said to > be contained in the union of all finite trees? Because an infinite path is a union of paths. And when you construct unions of *sets* of paths you are not constructing unions of paths. > > > If all natural numbers are finite, their union as defined for > > > pathlengths cannot be infinite. > > > > Why not? > > Because there cannot be more than infinitely many natural numbers. All > those infinitely many numbers, however, are allegedly finite. And there are infinitely many such. And they are, indeed, all finite. The path lengths however do not map to natural numbers, which you assert. That is valid for finite paths, but not for infinite paths. Let's, for once, go to a proper definition (assuming all are ordered sets): given a path {n_1, n_2, ..., n_k} then the path length is |{n_1, n_2, ..., n_k}| = k. Clearly for finite paths the path length is finite, and so a natural number. However, when we construct the path: p = {n_1} U {n_1, n_2} U {n_1, n_2, n_3}, ... we get as path: {n_1, n_2, n_3, n_4, ...} the path length in this case is *not* a natural number. It is the cardinality of N. > > > Except you assert that an infinite union requires a number of infinite > > > size. > > > > It is not a natural number. > > So it cannot be in the set of natural numbers. Right. > > That number is not an element of a tree. The elements of the tree are > > *nodes* (by your own definition). The union of the finite trees contains > > finite elements only (i.e. nodes at a finite distance from the root). > > But there are infinite sequences of such elements. > > Don't you accept that a finite tree simutaneously represents a finite > chain of nodes? And that an infinite tree represents an infinite chain > of nodes? Oh, it does contain them as subsets, yes. > > > > And, as > > > > your P(i) are sets with finite elements, their union does not contain > > > > an infinite element. On the other hand, their union *is* an infinite > > > > set. And so it is an infinite set of finite elements. > > > > > > The cardinal number is infinite, each pathlength is finite. > > > > Right. > > But you said that there is an infinite pathlength corresponding to an > infinite set of paths. But that set of paths is *not* the union of the P(i). Why do you think that the union of sets with finite elements does contain an infinite element? > > > What was your argument distinguishing infinite unions of finite trees > > > and finite paths, respectively? > > > > None. I distinguish unions of finite trees and unions of *sets* of > > finite paths. Above you are using P(1) U P(2) U ..., where each > > P(i) is a *set* of finite paths. > > What about unions of sets of finite trees? Would they work completely > different from unions of sets of paths? In particular, would the union > of sets of finite trees not contain or establish the inifnite tree? The union of sets of finite trees would neither contain, nor establish, an infinite tree. It would not contain one, because none of the constituent sets contains an infinite tree. It would not establish one, because the union is a set of trees, not a tree. A tree is a set of nodes, a set of trees is a set of sets of nodes. > > > > Wrong. Infinite paths do not have a path-length that is a > > > > natural number. > > > > > > Pathlengths ARE natural numbers. > > > > For finite paths. > > In all cases, by definition. *What* definition? -- 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 1 Feb 2007 21:18 In article <kAau6RggoiwFFwKO(a)phoenixsystems.demon.co.uk> Andy Smith <Andy(a)phoenixsystems.co.uk> writes: .... > 1) Is there a set with cardinality greater than N but less than R ? The Continuum Hypothesis states that there isn't, but it is not provable. And so some people use it as an axiom, others do not. > 2) I am (dimly aware) that mathematicians talk about an infinite range > of cardinalities. Is that just an abstract concept, or can you point to > some set and say e.g. that has a higher cardinality than the reals? That has already been shown by Cantor. In one of his proofs he did show (in effect) that given a set K, the set of subsets of K has a cardinality that is strictly larger than the cardinality of K itself. The most succinct proof of this is due to Hessenberg. The proof is fairly simple. Consider a set K and its powerset (i.e. set of subsets) P(k). Assume that there is an injection f: K -> P(K) (those clearly do exist, consider f(k) = {k}). Now we set out to prove that f (as given) can not be a surjection. Clearly for each 'k' in K, f(k) is a subset of K, so we can consider those 'k' where 'k' is an element of f(k) and those where 'k' is not an element of f(k). That defines two subsets of K, call them Y (the 'k' in that set map to subsets that contain 'k') and N. Now the question is, is there a 'k' that maps to N? (1) Assume 'k' in N, but by the definition of N, the elements do not map to a subset that contains themselve, so 'k' not in N. (2) Assume 'k' in Y, but by the definition of Y, the elements do map to a subset that contains themselve, and N does not contain 'k', so 'k' can not be in Y. As 'k' is in neither Y nor N, there is no 'k' that maps to N and so the map is not surjective. -- 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 1 Feb 2007 21:27
In article <JlH1jWkWEmwFFwNj(a)phoenixsystems.demon.co.uk> Andy Smith <Andy(a)phoenixsystems.co.uk> writes: .... > Case 2: a variant on that, we have colours black and white, and start > off with the interval 0-1 painted white. On each iteration, each > interval is subdivided, with the first sub-segment of each sub-interval > painted black if its parent interval was white, or vice-versa. Then, > after an infinite number of iterations, is each real point painted black > or white? And, if we decide to delete all white points after an infinite > number of iterations, would that make any difference? Some numbers will change between black and white indefinitely. So you can not ask whether there is an end colour. This is quite similar to the physical question about a fly going from train to train to an imminent disaster (head on collision). The question not being how many miles the fly did travel, but what direction the fly did travel when the disaster did occur. > Or is that all completely irrelevant? Not irrelevant, but unanswerable. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |