Cantor Confusion
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From: Virgil on 13 Dec 2006 15:12 In article <1166025842.895194.76700(a)16g2000cwy.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > > > A_1 = {1} > > > > > > A_2 = {1,2} > > > > > > A_3 = {1,2,3} > > > > > > > > > > > > B = {1,2,3} > > > > > > > > > > > > then B is contained in the last A_i. If there is no last A_I, then > > > > > > there is > > > > > > no A_i that contains B > > > > > > > > > > That has nothing to do with "last". > > > > > > > > If A_i contains B, then A_i contains any A_j. > > > > Therefore A_i is "last". > > > > > > > > No comment? > > If B contains any A_i then B is the last. That assumes, without evidence, that there is last. If WM choses to assume that every set is finite, as he has been doing, then he is working in a different system than everyone else, and what he has to say need only be valid within his personal system, and not in anyone else's, at least unless they choose to accept all of his assumptions. Which most of us do not.
From: Virgil on 13 Dec 2006 15:15 In article <1166026029.664903.246630(a)79g2000cws.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > > >> Mind and brain and representation of (abstract) entities therein is > > >> still off topic in sci.math. > > > > > > This decision is > > > 1) wrong > > > 2) not yours. > > > > Which _mathematical_ institution is doing research in the field of > > "representation of (abstract) entities in mind or brain"? > > New aspects are new because they have not been treated yet. "Aspects", new or old, are non-pure-mathematical when they become tied to physical realities. Applications of mathematics is a different field.
From: Virgil on 13 Dec 2006 15:28 In article <1166035958.305158.282870(a)79g2000cws.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > > > > That one which goes left from the root is not engaged, because we > > > > > need > > > > > only half of the set of edges. > > > > > > > > So your construction is not a surjection. > > > > > > Of course it is! It is a surjection from the set of edges onto the set > > > of paths. > > > > Than again. To what number maps the edge that goes left from the root? > > If it is a surjection, there should be one. > > > A surjection onto the paths covers all elements of the *range* = every > edge. It is not necessary that every edge is mapped on a path, to show > that there are not less edges than paths. It is only necessary that > there is no path without edge mapped on it. Which necessity WM has not accomplished. All Wm has done is to map infinite sequences of edges onto paths, but that is trivial since an infinite sequence of edges is essentially what a path is. > > You had said that you had constructed a surjection. A surjection is a > > mapping with specific properties. It is necessary to know what a mapping > > is in order to construct a surjection, which you did not do. And I have > > *no* idea what you mean with the second half of your sentence. > > If someone makes an asserted list of all real numbers and hands it out > to Cantor, then Cantor can construct a diagonal number which is not in > the list, contradicting the assertion. > > If someone makes a binary tree of all real numbers and hands it out to > Cantor, then Cantor cannot construct a diagonal number which is not in > the tree, so no contradiction of completeness is possible. Actually anyone can. One does it by dealing with successive pairs of binary digits to make the representation equivalent to a base 4 representation, in which the cantor diagonal method works quite nicely. > > The problem with Cantor's theorem aleph0 < 2^aleph0 is that the > function f(x) = 2^x must be discontinuous: Since the domain and codomain of that function are neither of them topological spaces, or even metric spaces, the notion of continuity is impossible to consider. > By means of the binary tree I can exclude any discontinuity. What is your topology or metric on the binary tree? Absent metrics, or at least a topologies, one cannot even speak of continuity or discontinuity of a function.
From: Virgil on 13 Dec 2006 15:39 In article <1166036620.491152.139940(a)n67g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > Can't see any support for your claim, that "functions cannot grow". > > In *set theory* a set does not grow. In set theory a function is a set. > In set theory a function cannot grow. > In my view a function can grow. A function can be abbreviated as number > variable or as variable number. A function as a set is fixed and does not grow, but if the domain and codomain of the function are ordered sets, as for example with real functions, the the function can be an increasing function. E.g., f " R --> R " x |--> x^3 is an increasing real function because If a in R and b in R with a < b then a^3 < b^3. In this sense a real function can "grow". > > > I am glad to hear that many people read my texts. Repeated reading > > > supports understanding. Repetition is a Baustein of learning. > > > > It is you who repeats. > > Of course, I do so in order to support your understanding. It clarifies our understanding of your errors when you repeat them. > > > Are you still not convinced of your mathematical > > revisionism? > > You misread. It is called mathe-*realism* not mathe-revisionism. That depends on who is doing the calling. Mathematicians tend call it things like "math-revisionism". Only anti-mathematicians like WM call it by favorable names. > > Regards, WM
From: mueckenh on 13 Dec 2006 16:00
mueckenh(a)rz.fh-augsburg.de schrieb: KORREKTION A surjection onto the paths covers all elements of the *range* = every path. Regards, WM |