Cantor Confusion
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From: mueckenh on 7 Dec 2006 06:59 Dik T. Winter schrieb: > In article <1165421463.339178.48680(a)j44g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > William Hughes schrieb: > ... > > > Consider A= [0,1), the set of real > > > numbers in greater than or equal to 0 and less than 1. > > > This set is totally ordered. This set is > > > composed of finite elements. We have the true statement > > > > > > For every element r in A, there exists an element > > > s in A such that r < s. > > > > > > However, if we simple reverse the quantifiers we get > > > the false statement > > > > This set is not finite. > > The set in the quantifiers you are using is not finite either. The > quantifier are not over a single line, but over the set of natural > numbers. For finite natural numbers, we have finite lines only. It is not the question of a single line. Every line is finite. Therefore there is no line where quantifier reversal could not be applied. "The great infinite" cannot help you. That is why I devised the EIT. Regards, WM
From: mueckenh on 7 Dec 2006 07:06 Dik T. Winter schrieb: > In article <1165403079.475990.150660(a)f1g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > > > > > I have difficulty with the tree because your explanations are confused > > > > > and sometimes contradictionary. > > > > > > > > Which one? > > > > > > Many. > > > > Care to name one? > > I disremember, and it is not easy to find among the many articles by you. > > > Please do not mistake your misunderstandings for > > errors of mine. For instance I never stated that nodes represent > > numbers, as you erroneously believed. > > Pray reread what I wrote: "the nodes can be made to represent numbers in > your tree". That is an easy exercise, I even did show it. The same for > the edges, I did show that too. So actually the nodes and edges also > represent numbers in some way. In the same way as the first few digits of a real number represents a number. 3.1, 3.14, and so on represent numbers in some way. But that is not at all important or interesting for the tree argument. > > > > > > You state that you are using limits with your infinite paths? > > > > > > > > Of course. The paths are nothing else but another way of denoting a > > > > real number in binary representation. > > > > > > But in that case you are doing something non-standard. > > > > Not at all! I represent numbers by standard binary notations. > > It is using the limits where you are doing something non-standard. I do nothing. The tree cares that even in the limit the number of paths cannot become uncountable. 2^n remains the cardinal number of a countale set, even in the limit n --> oo. That's why I devised the tree! Regards, WM
From: mueckenh on 7 Dec 2006 07:08 Dik T. Winter schrieb: > In article <1165403177.893469.76590(a)j72g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > ... > > > > To have a big mouth is not enough to create a world, not even a notion. > > > > You would see that if you tried to say where the assumed object > > > > existed. > > > > > > In my mind. I can reasonably think about the set of all natural numbers. > > > Honest, I have no problem with it. > > > > You believe you could. That is a difference to "can". Other people > > think they can reasonably think of having an immortal soul. > > How do you know I can not? How do you know what I can think? Perhaps you think so. But facts are contrary. Just as in case of infinity. Regards, WM
From: Han de Bruijn on 7 Dec 2006 07:15 Franziska Neugebauer wrote: > I don't know any mathematician who "pretends" an ability of > "knowing" (what does this mean?) every integer. Who "pretends" so? Yeah, sure .. Everybody on earth knows what "knowing" means in common conversation, except mathematicians. Han de Bruijn
From: mueckenh on 7 Dec 2006 07:18
Dik T. Winter schrieb: > Apparently you do not allow that a term can denote different things in > different realms. Mathematical reality is that it *can*. That's all depening on the definition. I advocate the idea that sets can change and numbers can grow because the common (contrary) definition leads the direct way to the (false) result of actuallity we observe in modern set theory. There is no room for potential infinty. But I am not surprised to hear this harsh critic. With their words and with their notions modern set theory will perish. > > > Can you provide a quote from Hrbacek and Jech where they state that > everything is a set? > -- Pae 2: In this book, we want to develop the theory of sets as a foundation for other mathematical disciplines. Therefore, we are not concerned with sets of people or molecules, but only with sets of mathematical objects, such as numbers, points of space, functions, or sets. Actually, these first three concepts can be defined in set theory as sets with particular properties, and we do that in the following chapters. So the only objects with which we are concerned from now on are sets. But this point of view is also entertained in many other modern books: In ZFC everything is a set. Regards, WM |