An uncountable countable set
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From: mueckenh on 13 Jul 2006 07:39 Virgil schrieb: > In article <1152736018.461492.108050(a)p79g2000cwp.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > The binary tree: All paths, not yet disperged, start through the root > > edge a. > There is no need for a root edge at all. the tree can quite comforably > be rooted in a rot node from which two edges branch out to two cild > nodes, and so on. > > > > (Let us denote them as a single path as long as they are > > together.) > > As paths are sets of edges, this does not work. Call it "bunch of paths cross sections", if you like. I abbreviate that by path. > > Map this root edge a on this single path. In the next level > > the path splits in two paths. Map half of edge a on each of them. The > > right one passing through edge b, gets b mapped on it and it inherits > > half of a. After splitting again, each of the paths gets the next edge, > > say c, half of b and quarter of a. > > > > | a > > o > > / \ b > > o o > > / \ / \c > > This tree contains 4 paths indicated by sequences of two branchings, > left or right right, so that {LL,LR, RL, RR} represents the set of all > paths for the tree as shown. > > For larger binary trees one gets more and longer strings of left to > right branchings. > > For infinite binary trees each such list of branchings is an infinite > sequence. And there are uncountably many possible such infinite > sequences. But we know that there are nearly twice as much edges. There are infinitely many of them too. And the the set of edges is countable. Regards, WM
From: mueckenh on 13 Jul 2006 07:41 Virgil schrieb: > In article <1152638895.480011.23630(a)h48g2000cwc.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > > But not all unary sequences are representations of natural numbers. > > > 0,111... is a sequence which does not represent a natural. > > > > And it cannot be completely indexed by natural numbers. > > That is precisely wrong, it is only the set of all natural numbers which > CAN index it. It cannot be completely indexed by all natural numbers. That is no question. Therefore the set of all does not exists. Regards, WM
From: mueckenh on 13 Jul 2006 07:43 Virgil schrieb: > 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 Both do no exist. That is the reason why Cantor's arguing is correct and mine is not. I see. Regards, WM
From: mueckenh on 13 Jul 2006 07:45 Virgil schrieb: > > > > But it does not guarantee what is existing there. In order to find out > > you must count. > > I do not know what "axiom of infinity", 'mueckenh" is referring to , but > the ones in ZF, or ZFC or NBG guarantee existence of all of the > "naturals" ( as finite ordinals) without having to count anything. Perhaps those naturals guaranteed by the axiom even cannot be used for counting? They are, after all, no numbers. Regards, WM
From: mueckenh on 13 Jul 2006 07:49
Virgil schrieb: > In article <1152718980.956031.285220(a)75g2000cwc.googlegroups.com>, > 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. > > Actually a unary numeral, or any other numeral for a natural number, can > only index one position. The MEMBERS of a natural, however expressed, > can index up to that natural. The members of N can index all natural > positions. Why then is 0.111... not in the sequence 0.1, 0.11, 0.111, ...? 0.111... cannot be indexed by the members of any natural. Does 0.111... have more 1's than can be indexed by any natural? Or what prevents it from being indexed by any of the naturals? Regards, WM |