Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: mueckenh on 4 Dec 2006 08:09 William Hughes schrieb: > We extend this to potentially infinite sets: > > A function from the set potentially infinite set A to the > potentially infinite set B is a potentially infinite set of > ordered pairs (a,b) such that a is an element of A and b is > an element of B. > > We can now define bijections on potentially infinite sets > and extend the bijection equivalence relation to include > potentially infinite sets. Thus we can define > equivalence classes under bijection of potentially infinite sets. > Thus we can define "cardinal numbers" of potentially > infinite sets. > There is only one "cardinal number". In order to apply any of Cantor's proofs of higher cardinal numbers, a set of aleph_0 must be complete. But it cannot be complete in potential infinity. Regards, WM
From: Franziska Neugebauer on 4 Dec 2006 08:22 mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: [...] > Everybody knows what the number of ther EC states is. Everybody except you knows that EC states are not part of any set theory. [...] >> > > You are, of course, entitled to use another definition, but that >> > > will not clarify the discussion at all (and you are not using >> > > standard set theory). >> > >> > Hrbacek and Jech teach standard set theory including the fact that >> > in ZF everything is a set. >> >> Yes? > > I remember that you opposed. Now you agree because you have learnt? Not everything is "in" ZF. Neither the EC nor their states are "in" ZF. Hence not everything you know is necessarily a set. >>Do they define sets as allowed to grow? Not in the quote you supply. >> There they talk about set valued variables that can grow. > > No. X and Y do not grow, they remain "X" and "Y". The set they denote > does grow. "To grow" colloquially means "1. To increase in size by a natural process." "Process" means "1. A series of actions, changes, or functions bringing about a result" (source: thefreedictionary) Usually the "series of action" means a succession in _time_. Since mathematics does not comprise a (physical) concept of time we have to define what "to grow" shall mean mathematically. Here is my proposal: Definition: A _function_ f: A |-> B /grows/ iff there exist a1 < a2 of dom(f) and f(a1) < f(a2). We use the abbreviation "f grows" for of "the function f grows". A set (value) denoted by the symbol "X" is refered to by writing X. If we say X grows. This means according to my defintion: 1. X is a function, 2. There exist two values x1, x2 of dom(X) with x1 < x2, and 3. X(x1) < X(x2) If we assign, for example, the set { 0, 1, 2 } to the variable X we _cannot_ meaningfully say that "X grows" because { 0, 1, 2 } is not a function. > The number of EC states may be n. "n" does not grow. The > number denoted by "n" does grow. Set theoretically numbers are sets. Only sets which are functions can grow. To model the "number of states of the EC" set theoretically we must define the number-of-states-of-the-ec-function n. This function should contain the ordered pairs (1958, 6) and (2006, 25). According to my definition n _does_ grow. NB the numbers 6, 25, 1958 and 2006 do not grow. F. N. -- xyz
From: mueckenh on 4 Dec 2006 08:26 Virgil schrieb: > > > Extending the concept of bijection from sets to potentially > > > infinite sets is trivial. > > > > May be if you apply your personal definition of potentially infinity, > > but not if you apply the generally accepted definition. > > What "generally accepted" meaning is that? Most mathematicians do not > accept that a set can be "potentially" infinite without being actually > so. Most "mathematicians" even don't know what potentially infinite is. And if they think they know, they mistake the one for the other: Trotz wesentlicher Verschiedenheit der Begriffe des potentialen und aktualen Unendlichen, indem ersteres eine veränderliche endliche, über alle Grenzen hinaus wachsende Größe, letztere ein in sich festes, konstantes, jedoch jenseits aller endlichen Größen liegendes Quantum bedeutet, tritt doch leider nur zu oft der Fall ein, daß das eine mit dem andern verwechselt wird. [G. Cantor, Gesammelte Anhandlungen, p. 374]. > > > > The problem is although the term "finite" is used in the > > > definition of A, A does not have many of the properties > > > that are usually associated with finite sets. In particular > > > it is not possible to produce A by induction. > > > > A potentially infinite set like N can be produced by induction. > > If it is a set, it is not merely "potentially" anything, and if it is > merely potentially something then it is not a set. You do erroneously believe that you define the properties of a set. > > > > An > > actually infinite set cannot be produced by induction nor can it be > > produced by other means because it is simply nonsense. > > That may be true in WM's as yet unspecified, and possibly unspecifyable, > axiom system, but is false in ZF or NBG or NF. The impossibility of producing an actually infinite set by induction should be obvious. > > > > > > The only way of > > getting it is to postulate its existence by an axiom, although the > > degree of nonsense is not reduced by this method. > > The only way of getting anything in mathematics is by postulating it > either directly or by postulating the means of constructing it. No. That is only necessary for non-existing entities ("existing" is here to be understood as existence and not as belief). > > > > > Try to learn the commonly accepted meaning of "potentially infinite". > > There is no such thing as a 'commonly accepted meaning of "potentially > infinite"'. > > > I > > am not wiling to discuss your personal definitions. > > We do not wish to discuss your personal definitions either, but you keep > thrusting them at us. Why don't you stop? Regards, WM
From: Franziska Neugebauer on 4 Dec 2006 08:28 mueckenh(a)rz.fh-augsburg.de wrote: > So you can construct the set of all real numbers (of the interval [0, > 1] in binary representation) by: > > 0.0 > 0.1 > 0.01 > 0.11 > ... > > This set is countable. 1/3 is missing. F. N. -- xyz
From: Franziska Neugebauer on 4 Dec 2006 08:32
mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: [...] >> We do not wish to discuss your personal definitions either, but you >> keep thrusting them at us. > > Why don't you stop? You are so amazingly amusing. F. N. -- xyz |