Cantor Confusion
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From: mueckenh on 5 Jan 2007 09:01 Virgil schrieb: > In article <1167860527.440727.143820(a)i80g2000cwc.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Dik T. Winter schrieb: > > > > > > > > But you did not say which edges or nodes distinguish the complete tree > > > > from the union of all rational trees. > > > > Do you believe that there are such distinguishing edges but that one > > > > cannot name them? > > > > Or do you think that there are same edges in both trees but that some > > > > paths can form in one tree which cannot from in the other tree? > > > > > > The last one. In the union there is *no* infinite path. > > > > > > > Wouldn't both answers point to some matheology? > > > > > > No, they are based on the definition of union and some elementary logic. > > > (1) Each node is in one of the finite trees, so it is also in the union. > > > > So there is an infinite number of nodes in the union of finite numbers > > of nodes. > > Only if WM can prove that one can generate the necessary infinite number > of trees from a merely finite number of nodes. Which I doubt. So the union of all finite trees has ony a finite number of nodes? > > > > > (2) Each edge is in one of the finite trees, so it is also in the union. > > > > So there is an infinite number of edges in the union of finite numbers > > of nodes. > > Only if WM can prove that one can generate the necessary infinite number > of trees from a merely finite number of edges. Which I doubt. So the union of all finite trees has ony a finite number of edges? > > > > > (3) Infinite paths are in none of the finite trees, so they are also not > > in the union. > > > > So the set of natural numbers is not in the union of all initial > > segments of natural numbers? > > Certainly not as a member. But it is the very set which has AS MEMBERS > all of the members of all of the initial sets. And the union is the infinite set N? > > > > > > With which of the above three statements do you disagree? > > > If (1), name a node not in one of the finite trees. > > > If (2), name an edge not in one of the finite trees. > > > If (3), name an infinite path that is in a finite tree. > > > Apparently you agree with all three, but at the same time states that > > > the completed infinite tree (which has infinite paths) is the same as > > > the union of the finite trees. > > > > That is wrong. If an infinite number of nodes is in any path of the > > union, then the path is infinite too. > > But it is impossible for any finite path in any finite tree to contain > more than finitely many nodes, or edges. Fine. That is my arguing with respect to the natural numbers. Their number is not finite. > As no path in any set of the > union has infinitely many nodes, and the union can contain only those > paths, from which finite tree is WM getting his infinite path? Fine. Infinity does not exist. (Because the complete tree cannot contain more than the union of all finite trees.) > > > > > If EVERY edge of a path is shared by another path, then both paths > > cannot be distinguished. > > There is no "both paths" until one has chosen both paths to be > compared, at which point for those two one edge in each not in the other > is easy to find, the two edges branching from last node they have in > common. There is a both paths, if you claim that no edge is unique. Of course it is not your "both paths" which are different by your choice and definition. Regards, WM
From: mueckenh on 5 Jan 2007 09:04 Virgil schrieb: > In article <1167860661.713987.282460(a)k21g2000cwa.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > If EVERY edge of a path is shared by another path, then both paths > > cannot be distinguished. > > That is only true in finite trees. In an infinite tree, given any path > and any finite n, there is anoth path sharing the first n edges but > separating after n edges. We are looking for two paths sharing every edge. (Otherwise there must be an edge not shared by two ore more paths.)> > > The reason is that I do not believe in differen trees which are > > identical > > Except when YOU want them to be. I simply accept that identity with respect to all properties means identity with respect to all properties. Regards, WM
From: William Hughes on 5 Jan 2007 09:04 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > I do not assume that 0.111... exists. I only assume that > > given that 0.111...1 exists you can show that 0.111...11 exists. > > > That position is correct. It is called potential infinity. > > > > > > That is the same with the lines. Why should the diagonal exist actually > > > but the system of lines should not exist actually? (1/9 = 0.111...) > > > > Indeed. It is the same with lines. It is always possible to find > > another line. However, iii says nothing about whether the > > diagonal "actally exists" (or equivalently whether the > > system of lines "acutally exists"). > > But you always assert that the the complete diagonal exists, i.e., a > diagonal which could not be extended. Then you must also accept a line > system which cannot be extendend. No. At no time do I assume that the complete diagonal exists. > > > > > > > iv: Not all the elements of the diagonal exist > > > > > > Not all natural numbers in unary representation 0.1, 0.11, 0.111, ... > > > exist. If all elements of the diagonal exist (which are the last digits > > > of the unary numbers) then these unary numbers must exist too, as far > > > as I understand existence. > > > > Any discussion of iv is completely irrelevant. iv is neither needed > > nor used. > > Then drop it. I have never used it ("neither needed nor used"). > > You used to use ~iv: all the elements of the diagonal exist > Only to point out that I neither need nor use the assumption ~iv. > > The contradiction is: > > > > A: L_D must have the property that given any set of elements > > that exist in L_D one can always find another element of > > L_D > > [This follows immediately from i ii and iii] > > It is not different for the lines. > > > > B: L_D has a largest element. [this follows from the fact the > > L_D > > is a line] > > This follows from the assumption (iv) of complete existence of L_D, > which implies complete existence of the system of lines. If you drop > it, then the contradiction vanishes. No, it follows from the fact that L_D is a line. A line has a largest element. L_D is not the system of lines. No assumption about the complete existence of the system of lines is needed or used. - William Hughes
From: mueckenh on 5 Jan 2007 09:11 Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > Franziska Neugebauer schrieb: > > > > It is without any value to follow your text. > > LOL. Head-in-the-sand? Lack of interest in useless sophisms. If you are interested in meaningful discussion, then try to find out, for instance, the difference between the complete binary tree (with irrational numbers) and the union of all finite binary trees (without such representations). Your party is going down. Regards, WM
From: mueckenh on 5 Jan 2007 09:16
William Hughes schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > > > I do not assume that 0.111... exists. I only assume that > > > given that 0.111...1 exists you can show that 0.111...11 exists. > > > > > That position is correct. It is called potential infinity. > > > > > > > > That is the same with the lines. Why should the diagonal exist actually > > > > but the system of lines should not exist actually? (1/9 = 0.111...) > > > > > > Indeed. It is the same with lines. It is always possible to find > > > another line. However, iii says nothing about whether the > > > diagonal "actally exists" (or equivalently whether the > > > system of lines "acutally exists"). > > > > But you always assert that the the complete diagonal exists, i.e., a > > diagonal which could not be extended. Then you must also accept a line > > system which cannot be extendend. > > No. At no time do I assume that the complete diagonal exists. > > > > > > > > > > > iv: Not all the elements of the diagonal exist > > > > > > > > Not all natural numbers in unary representation 0.1, 0.11, 0.111, ... > > > > exist. If all elements of the diagonal exist (which are the last digits > > > > of the unary numbers) then these unary numbers must exist too, as far > > > > as I understand existence. > > > > > > Any discussion of iv is completely irrelevant. iv is neither needed > > > nor used. > > > > Then drop it. > > I have never used it ("neither needed nor used"). > > > > > You used to use ~iv: all the elements of the diagonal exist > > > > Only to point out that I neither need nor use the > assumption ~iv. > > > > The contradiction is: > > > > > > A: L_D must have the property that given any set of elements > > > that exist in L_D one can always find another element of > > > L_D > > > [This follows immediately from i ii and iii] > > > > It is not different for the lines. > > > > > > B: L_D has a largest element. [this follows from the fact the > > > L_D > > > is a line] > > > > This follows from the assumption (iv) of complete existence of L_D, > > which implies complete existence of the system of lines. If you drop > > it, then the contradiction vanishes. > > No, it follows from the fact that L_D is a line. A line has a largest > element. L_D has a largest element? > L_D is not the system of lines. L_D consists of line ends, i.e., just of these largest elements. > No assumption about the complete existence of the > system of lines is needed or used. You use the complete existence of L_D. But perhaps you do not recognize it. Regards, WM |