Cantor Confusion
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From: mueckenh on 4 Dec 2006 09:48 Franziska Neugebauer schrieb: > 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. Of course. It is not a potentially infinte sequence. The above example was only posted to convince William Hughes of the necessity to have actually infinite sequences as real numbers. The section snipped by you explained this. Regards, WM
From: mueckenh on 4 Dec 2006 09:55 William Hughes schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > 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. > > You now agree that a potentially infinite set can have > a cardinal number and that this cardinal is not > a natural number. > I wrote: a "cardinal number". oo is not a cardinal number in the sense of set theory. > We have: there exists a bijection between sets or potentially infinite > sets > A and B iff the cardinal number of A is the same as > the cardinal number of B. > > Now apply this. > > The natural numbers form a potentially > infinite set. The diagonal contains the potentially infinite set > of natural numbers. There is nothing to contain! You are too much caught in the terms of set theory. You cannot have the complete set because then it would be complete, i.e., actually existing, i.e., actually infinite. > Therefore the cardinal of the set of elements of > the diagonal is not a natural number. For any line, the cardinal of > the elements of the line is a natural number. Therefore there > is no bijection between the diagonal and any line. > However, the line indexes contain the set of natural numbers > and form a potentially infinite set. There can be a bijection > from the diagonal to the line indexes. Compare Franziska's objection that 1/3 is not a member of my potentially infinite list of real numbers. Regards, WM
From: mueckenh on 4 Dec 2006 10:05 stephen(a)nomail.com schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > Dik T. Winter schrieb: > > >>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. The number of EC states may be n. "n" does not grow. The > > number denoted by "n" does grow. > > What do you mean by the 'the number denoted by "n" does grow'? > Currently the number of EC states is 25. In a month it will be 27. > Does that mean 25 is going to grow into 27? Will 25 no longer exist? > Or will 25 now mean 27? What do you mean by 'the number denoted by "25" does > grow'? It is only a matter of definition and in principle no reason for quarrel. But it is amusing to see ho set theorists insist on the complete and actual existence of the sets of numbers. Of course 25 will not switch to 27 but the number of states will switch from 25 to 27. That's all. Only by this notion we can talk of growing sets and introduce the notion of potential infinity. > > The idea that 25 is ever going to be anything but 25 is absolutely ridiculous. > The idea that a set ever changes is equally ridiculous. No. Compare Fraenkel et all. They talk about to look at the universe of all sets not as a fixed entity but as an entity capable of "growing". What they understand and how this growing can take place has lead to many misunderstandings by underinformed mathematicians. But however one may interpret their sentence. The universe of all sets can change, to put it cautiously. That is not at all ridiculous. Regards, WM
From: William Hughes on 4 Dec 2006 10:14 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > 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. > > > > You now agree that a potentially infinite set can have > > a cardinal number and that this cardinal is not > > a natural number. > > > I wrote: a "cardinal number". oo is not a cardinal number in the sense > of set theory. > > > We have: there exists a bijection between sets or potentially infinite > > sets > > A and B iff the cardinal number of A is the same as > > the cardinal number of B. > > > > Now apply this. > > > > The natural numbers form a potentially > > infinite set. The diagonal contains the potentially infinite set > > of natural numbers. > > There is nothing to contain! You are too much caught in the terms of > set theory. You cannot have the complete set because then it would be > complete, i.e., actually existing, i.e., actually infinite. If you want to avoid the word contain, reword "The diagonal contains the potentially infinite set of natural numbers." as "if x is an element of the potentially infinite set of natural numbers, then x is an element of the potentially infinite set of elements of the diagonal". The point remains that there can exist a bijection from the diagonal to the line indexes, but there cannot exist a bijection from the diagonal to any line. - William Hughes
From: Franziska Neugebauer on 4 Dec 2006 10:37
mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> mueckenh(a)rz.fh-augsburg.de wrote: >> > Dik T. Winter schrieb: >> [...] >> > Everybody knows what the number of the EC states is. >> >> Everybody except you knows that EC states are not part of any set >> theory. > > Oh, there are two of us. You forgot Cantor.Beitr�ge zur Begr�ndung > der transfiniten Mengenlehre. (That *is* a set theory). "Unter einer > "Menge" verstehen wir jede Zusammenfassung M von bestimmten > wohlunterschiedenen Objekten in unsrer Anschauung oder unseres Denkens > (welche die "Elemente" von M genannt werden) zu einem Ganzen." > Or do you insist on living creatures? Since "Cantor" is still present in the subject I have to ask you whether you want to discuss anachronisms or if you want to learn how contemporary set theory works. > Further I am in accordance with he sentence: "Sets are not objects of > the real world: they are created by our mind, not by our hands." Of > course I understand by EC states the mind-created set of EC states. I will not discuss this anachronisms but modern concepts instead: ,----[ <45742128$0$97220$892e7fe2(a)authen.yellow.readfreenews.net> ] | Not everything is "in" ZF. Neither the EC nor their states are "in" | ZF. Hence not everything you know is necessarily a set. `---- [Hilbert] >> 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. > > It is not unusual to say f(x) grows with x (if it does so). And "it does so" means by a definition: ,----[ <45742128$0$97220$892e7fe2(a)authen.yellow.readfreenews.net> ] | 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". `---- Not more, not less. Without a definition of "to grow" there is no meaningful notion of "to grow" in Z set theories (yet). >> 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. > > Correct. So the EC-states, after all, are in ZFC. Non sequitur. You simply apply set theoretical reasoning to some other field of interest. > I knew it. I know that you "know" many invalid things. The states of the EC are entities of a different universe of discourse and they are not "in ZFC". Linking up set theory with any other universe of discourse does not necessarily make the entities of the latter become members of the former. >> NB the numbers 6, 25, 1958 and 2006 do not grow. > > That is not a surprise. I agree completely with you. I would appreciate if you now reexamine your own statement ,----[ <1165228714.543673.34570(a)j72g2000cwa.googlegroups.com> ] | No. X and Y do not grow, they remain "X" and "Y". The set they denote | does grow. The number of EC states may be n. "n" does not grow. The | number denoted by "n" does grow. `---- I am especially interested in the growing number and the growing set. LOL F. N. -- xyz |