An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Virgil on 13 Jul 2006 04:37 In article <1152639432.511655.220040(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > > Except that after each (1,2), there will only be a finite initial > > subsequence in numerical order and an infinite terminal sequence not yet > > ordered. > > > > Using 1 origin indexing, the nth occurrence of (1,2) occurs at the > > (n^2 + n)/2 th position in the list of "transpostions". > > > > At which (n^2 + n)/2 th operation is the entire list ordered? > > At which number can I find the last element of Cantor's diagonal? The successor to the one which finishes your ordering by magnitude. Which, since neither exists, works correctly
From: mueckenh on 13 Jul 2006 06:03 Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Franziska Neugebauer schrieb: > >> > >> You are using "to index" simultaneously for two different meanings. > >> The first is "to index a specific position" the second is undefined. > >> What does "0.1111 is indexed by the list number 4 = 0.1111" exaclty > >> mean? > > > > The unary number n indexes every position 1,2,3,...,n. > > How exactly is that achieved? The unary number n provides the existence of every m with 0 < m =< n in my list. All these m have been used to index the digits of n, so these indexes, now in n, can serve to index further numbers. > >> What does "completely indexed" mean? > > > > All digits are indexed. > > All positions *are* indexed. There is a bijection between the index set > N and the figures a_i: > > ... i ... > ^ > | > v > ... a_i ... That is correct for all a_i of list numbers = naturla numbers. That is not possible for 0.111... . By the way, this is the very reason why 0.111... is not in the list. Regards, WM
From: mueckenh on 13 Jul 2006 06:16 David Hartley schrieb: > The transpositions are relevant in so far as WM claims that Cantor wrote > that a sequence of transpositions can not change the order-type. Either > Cantor was wrong (or using a different idea of the limit), or WM is > misinterpreting him, my German is not good enough to tell. Daraus folgt, daß solche Umformungen einer wohlgeordneten Menge die Anzahl derselben ungeändert lassen, welche sich auf eine endliche oder unendliche Folge von Transpositionen von je zwei Elementen zurückführen lassen, ... It follows that only such transformations of a well-ordered set leave its (ordinal) number unchanged, which can be derived from a finite or infinite sequence of transpositions of each two elements, ... > In any case, > it's a proposition WM needs to prove to complete his argument, and he > hasn't done so. What should that be good for? In the binary tree everyone can see that no path can split without two additional edges. So it is by no means possible that there were more paths than edges. Nevertheless the set theorists go through eyes wide shut and do not care. Who would accept logic arguments if set theory was concerned? Regards, WM
From: mueckenh on 13 Jul 2006 06:29 Dik T. Winter schrieb: > However, some say that Cantor has also written an article about the > uncountability of real numbers using decimal notation, but I have not > yet found it. I am starting to doubt that. I doubt that too, because neither in his collected works nor in his correspondence I have found any hint on that. > What I find is the following: > 1. Cantor's first diagonal argument: proves (amongst others) that the > rational numbers are countable. > 2. Cantor's second diagonal argument: proves that there are sets with > cardinality greater than the cardinality of the natural numbers (the > proof we are discussing here). > 3. His proof that the reals are not countable (published in 1874). > Of course, Zermelo did show that (2) can be transformed to a proof that > the reals are not countable, but I do not think that Cantor did do that > transformation. Cantor himself generalized his proof (in the same paper) to all well-defined sets like the real numbers (= Linearcontinuum) and functions which only can take the values 0 and 1. (Therefore I said one could safely interpret his list as a list of binary numbers. Zermelo only remarked the problem with double representation of binary fractions.) Cantor concluded: daß die Mächtigkeit wohldefinierter Mannigfaltigkeiten kein Maximum haben oder, was dasselbe ist, daß jeder gegebenen Mannigfaltigkeit L eine andere M an die Seite gestellt werden kann, welche von stärkerer Mächtigkeit ist als L. For non-German-readers: He recognized that cardinalities do not have a maximum. Regards, WM
From: Franziska Neugebauer on 13 Jul 2006 06:33
mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> > >> > Franziska Neugebauer schrieb: >> >> >> >> You are using "to index" simultaneously for two different >> >> meanings. The first is "to index a specific position" the second >> >> is undefined. What does "0.1111 is indexed by the list number 4 = >> >> 0.1111" exaclty mean? >> > >> > The unary number n indexes every position 1,2,3,...,n. >> >> How exactly is that achieved? > > The unary number n provides the existence of every m with 0 < m =< n > in my list. All these m have been used to index the digits of n, so > these indexes, now in n, can serve to index further numbers. Could you reword the last paragraph by means of coherent language? >> >> What does "completely indexed" mean? >> > >> > All digits are indexed. >> >> All positions *are* indexed. There is a bijection between the index >> set N and the figures a_i: >> >> ... i ... >> ^ >> | >> v >> ... a_i ... > > That is correct for all a_i of list numbers = naturla numbers. That is > not possible for 0.111... . Your presumption that infiite things "are not possible" has no meaning in contemporary mathematics. Do you still want to show a contradiction that is not based on your absurd presumptions but within i.e ZFC? F. N. -- xyz |