Cantor Confusion
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From: Virgil on 5 Jan 2007 16:34 In article <1168001536.774868.44790(a)q40g2000cwq.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > It is without any value to follow your text. > > 1) My "ideas" are identical with what you call irrational numbers. > 2) But these numbers are not numbers, because they cannot be > approximated to any given epsilon. > > I will not withdraw (1) or (2), and they do not constitute a > contradiction. > > Regards, WM Then reasonable people will agree that a WM-idea is really an irrational number. There is nothing in any standard definition of "number" about epsilons.
From: Virgil on 5 Jan 2007 16:43 In article <1168002324.590103.255240(a)s34g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1167859855.107241.239690(a)k21g2000cwa.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > Virgil schrieb: > > ... > > > > it is quite similar in that one has in both cases an infinite set of > > > > all > > > > whose members are finite. > > > > > > That is precisely the infinite string of finite digit indexes of > > > irrational numbers and similarly the infinite string of finite nodes of > > > infinite paths. But one has no infinite paths in the tree? > > > > Not in the union of finite trees, but it is in the union of the full tree. > > The two are not the same. > > Name a level or a node or and edge which are in one of the trees only. > Do not argue that paths can differ without any different node or edge. > That is too obvious a nonsense. That is your nonsense, not ours. We never have said that. What we have said is that a set of paths all of which end does not contain any paths which do not end. > But, according to your point of view, N exists as the union of all > finite sequences {1, 2, 3, ..., n}. Why do infinite paths not exist as > the union of all finite paths? WM is not taking the union of all finite paths but rather the union of all finite trees. And the only object that can be a member of that union of trees is an object which is a member of one of those trees. And every path in a finite tree is a finite path. So that WM cannot get where he wants to go by taking unions of finite trees.
From: Virgil on 5 Jan 2007 16:45 In article <1168004789.483025.179970(a)s80g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1167859668.778067.101400(a)i80g2000cwc.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Franziska Neugebauer schrieb: > > > > > > > > > > > Correct. x is an idea. It is identical to the entity which you call an > > > > > irrational number, i.e., an idea y. > > > > > > > > > >> Your answer to question 3 states > > > > >> > > > > >> x is different from y (due to some reason) > > > > >> <-> there is a difference between x and y (Q) > > > > > > Here is your error: > > > x is different from y' > > > <-> there is a difference between x and y' (Q) > > > i.e., there is a difference between x and what you believe is y. The > > > error lies in your assertion that y = y'. > > > > As there is only a y and no y', the error is entirely WM's by > > introducing what does not exist. > > > > "by introducing what does not exist." > > That is a wise sentence. Now you should try to find out who introduced > what does not exist (like some paths in a tree). It is clear in the above that WM is the culprit.
From: Virgil on 5 Jan 2007 16:49 In article <1168005290.530898.280240(a)q40g2000cwq.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > So the union U{n | n e N} = U{1,2,3,...,n} is not N? What is missing? > > > > In my book, U{n|n e N} is not equal to U{1,2,3,...,n}. > > What tells your book about the difference? As far as I am informed, two > different sets can be distinguished by at least one element. For all n e N, (n+1) e U{n | n e N} For all n e N, Not (n+1) e U{1,2,3,...,n} > > > And, in addition to any common sense, it appears that an "h" is missing > > from WM's question. > > Yes, *here* you are right. > > > > > (I do not ask for a member. I do not ask whether an infinite path is a > > > member. I ask whether it is in the union of finite trees.) > > > > What is the "it" that you are asking about? > > An infinite path. if no finite tree contains an infinite path as a member then no union of finite trees can contain one as a member.
From: Virgil on 5 Jan 2007 17:01
In article <1168005688.719735.164370(a)42g2000cwt.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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? WM may believe that, but that does not follow from anything that I said or from anything logical. The union of all finite trees does not contain any infinite path but it does contain infinitely many finite paths, as well as infinitely many nodes and infinitely many edges,. > > > 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.) That infinite union of finite trees contains infinitely many nodes and infinitely many edges in infinitely many finite paths, but no infinite paths. > > > > > > > > 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. Every edge is, in some sense, unique, but I have no idea in what respect WM is requiring some edge to be unique. In the sense that if two paths are distingishable there is some edge in one but not in the other, fine. For any two infinite paths there are infinitely many such "unique" edges, in one but not the other. |