Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: mueckenh on 14 Oct 2006 04:09 William Hughes schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > Dik T. Winter schrieb: > > > > > In article <1160647755.398538.36170(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > ... > > > > It is not > > > > contradictory to say that in a finite set of numbers there need not be > > > > a largest. > > > > > > It contradicts the definition of "finite set". But I know that you are > > > not interested in definitions. > > > > We know that a set of numbers consisting altogether of 100 bits cannot > > contain more than 100 numbers. Therefore the set is finite. The largest > > number of such a set cannot be determined, as far as I know. > > There is a big difference between saying we do not know what > the value of the largest element of a set is and saying that > a set does not have a largest element. The set discussed above does not have a largest element. Every method of representing its largest number can be surpassed by another one, I believe. Note also: In advanced branches of science we have recognized that some entities cannot exist, in particular such entities which in principle cannot be determined. In mathematics this process is about to occur over short or long. Regards, WM
From: mueckenh on 14 Oct 2006 04:12 David Marcus schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > David Marcus schrieb: > > > > > > I am sure you are able to translate brief notions like "to enter, to > > > > > > escape" etc. by yourself into terms of increasing or decreasing values > > > > > > of variables of sets, if this seems necessary to you. Here, without > > > > > > being in possession of suitable symbols, it would become a bit tedious. > > > > > > > > > > Yes, I can translate it myself. However, that would only tell me how I > > > > > interpret the problem. > > > > > > > > Hasn't it become clear by the discussion? > > > > > > > > I use two variables for sequences of sets. Further I use a function. I > > > > use the natural numbers t to denote the index number. The balls are > > > > simply the natural numbers. I speak of "balls" in order to not > > > > intermingle these numbers with the index-numbers. > > > > > > > > The set of balls having entered the vase may be denoted by X(t). > > > > So we have the mathematical definition: > > > > X(1) = {1,2,3,...10}, X(2) = {11,12,13,...,20}, ... with UX = N > > > > There is a bijection between t and X(t). > > > > > > t is a number and X(t) is a set. If t = 1, then your sentence says, > > > "There is a bijection between 1 and X(1)". But, X(1) = {1,2,3,...10}. > > > So, I don't follow. What do you mean, please? > > > > That what is written. There is a bijection between the set of all > > numbers t and the set of all sets X(t). 1 is mapped on X(1), 2 is > > mapped on X(2), and so on. Is there anythng wrong? > > > > > > > The set of balls having left the vase is described by Y(t). So we have > > > > the mathematical definition: > > > > Y(1) = 1, Y(2) = {1,2}, ... with UY = N > > > > There is a bijection between t and Y(t). > > > > Here we have the same as above with Y instead of X. > > > > > > > > And the cardinal number of the set of balls remaining in the vase is > > > > Z(t). So we have the mathematical definition: > > > > Z(t) = 9t with Z(t) > 0 for every t > 0. > > > > There is a bijection between t and Z(t). > > Sorry, but I'm not following. I asked you for a translation into > Mathematics of the ball and vase problem. The problem in English ends > with a question mark. I don't see the question mark in your translation > above. Would you please state just the problem in both English and > Mathematics? Sorry, I don't know how to state the question " Hasn't it become clear by the discussion?" in what you think is mathematics. By the way: Every means to draw conclusions and to calculate results is mathematics. There is no need to prefer a certain language (unless there is someone who cannot speak another one). Would you assert Archimedes did not do mathematics, because he used only the Greek language and had not yet special symbols but Greek letters to denote numbers? Regards, WM
From: mueckenh on 14 Oct 2006 04:17 David Marcus schrieb: > Tony Orlow wrote: > > Dik T. Winter wrote: > > > In article <452d14fe$1(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > > > > Dik T. Winter wrote: > > > > > In article <1160551520.221069.224390(a)m73g2000cwd.googlegroups.com> "Albrecht" <albstorz(a)gmx.de> writes: > > > ... > > > > > > What is the difference to the diagonal argument by Cantor? > > > > > > > > > > That a (to the right after a decimal point) infinite string of decimal > > > > > digits defines a real number, but that a (to the left) infinite string > > > > > of decimal digits does not define a natural number. > > > > > > > > It defines something. What do you call that? If the value up to and > > > > including every digit is finite, how can the string represetn anything > > > > but a finite value? > > > > > > I define it as a string of digits and it does not represent a number. It is > > > only when you give proper definitions of what strings extending infinitely > > > far away to the left represent, that you can talk about what it represents. > > > In common mathematics there is no such definition. > > > > When Peano defines the natural numbers, does he talk about what they > > represent, or only how they are generated? > > If you are asking what Peano himself did, I don't know, since I'm not a > historian. > > The Peano axioms for the natural numbers are a bunch of axioms, not a > construction of the natural numbers. You can simply start with the > axioms and go from there or you can start with something more basic > (sets) and construct the natural numbers. If you do the latter, then you > prove that what you constructed satisfies the Peano axioms. In this > case, the axioms are theorems. Once you get to this point, you keep > going as in the first case. > > (To those who've had a course in mathematical logic: I'm aware that the > preceding ignores the question of what first-order language the Peano > axioms are stated in. You can assume I'm just using whatever language > I'm using for all my Mathematics.) ((And to those who consider taking a second course in mathematical logic: What are the foundations of logic? Where are they obtained from? Is there a theory which yields the axioms of logic as theorems?)) Regards, WM
From: mueckenh on 14 Oct 2006 04:18 Virgil schrieb: > In article <1160640069.503756.100380(a)e3g2000cwe.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Dik T. Winter schrieb: > > > > > In article <1160551520.221069.224390(a)m73g2000cwd.googlegroups.com> > > > "Albrecht" <albstorz(a)gmx.de> writes: > > > > David Marcus schrieb: > > > ... > > > > > I don't follow. How do you know that the procedure that you gave > > > > > actually "defines/constructs" a natural number d? It seems that you > > > > > keep > > > > > adding more and more digits to the number that you are constructing. > > > > > > > > What is the difference to the diagonal argument by Cantor? > > > > > > That a (to the right after a decimal point) infinite string of decimal > > > digits defines a real number, but that a (to the left) infinite string > > > of decimal digits does not define a natural number. > > > > And why is this so? Because an infinite string of digits is not at all > > defined. Only by the factors 10^(-n) this is veiled. The due digits > > become more and more unimportant because their contributions to the > > number size are pulled down by the increasing exponents. But this has > > been forgotten by Cantor whose diagonal proof attaches the same weight > > to every digit. That is obviously wrong. > > Cantor merely assumes, as do most mathematicians, that in mathematics, > as contrasted with physics, there need not ever be a last significant > digit in a decimal expansion. But this assumption is wrong as we obtain from the fact that elimination of all factors 10^(-n) leads to undefined results. Regards, WM
From: mueckenh on 14 Oct 2006 04:21
David Marcus schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > Please state an internal contradiction of set theory. Please use the > > > standard language of set theory/mathematics so that we can understand > > > what the contradiction is without needing to ask what all the words > > > mean. > > > > Good heavens, there are so many. Where shall I start with? > > > Consider the binary tree which has (no finite paths but only) infinite > > paths representing the real numbers between 0 and 1. The edges (like a, > > b, and c below) connect the nodes, i.e., the binary digits. The set of > > edges is countable, because we can enumerate them > > > > 0. > > /a\ > > 0 1 > > /b\c /\ > > 0 1 0 1 > > ............. > > > > Now we set up a relation between paths and edges. Relate edge a to all > > paths which begin with 0.0. Relate edge b to all paths which begin with > > 0.00 and relate edge c to all paths which begin with 0.01. Half of edge > > a is inherited by all paths which begin with 0.00, the other half of > > edge a is inherited by all paths which begin with 0.01. Continuing in > > this manner in infinity, we see that every single infinite path is > > related to 1 + 1/2 + 1/ 4 + ... = 2 edges, which are not related to any > > other path. > > Are you using "relation" in its mathematical sense? Of course. But instead of whole elements, I consider fractions. That is new but neither undefined nor wrong. > > Please define your terms "half an edge" and "inherited". I can't believe that you are unable to understand what "half" or "inherited" means. I rather believe you don't want to understand it. Therefore an explanation will not help much. Regards, WM |