Cantor Confusion
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From: David Marcus on 23 Jan 2007 13:48 mueckenh(a)rz.fh-augsburg.de wrote: > It is very simple to see that the set {2,4,6,...2n} always, i.e., for > every n, contains larger number than its cardinal number n. OK. > It is > impossible that the "whole" set of even numbers has a cardinal number > which is larger than every even number. Why? -- David Marcus
From: David Marcus on 23 Jan 2007 13:54 mueckenh(a)rz.fh-augsburg.de wrote: > Sorry, that IS my argument. The union of all finite trees is not a > finite tree. Therefore it has no finite paths. > But also: All finite trees contain only finite paths. Therefore the > union of all finite trees contains no infinite paths. What about the path in the infinite tree that always goes left? Are you saying it isn't a path or it isn't infinite? -- David Marcus
From: Virgil on 23 Jan 2007 13:54 In article <1169545581.873125.47200(a)q2g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Franziska Neugebauer schrieb: > > >> >> By induction you only define (if at all) every finite union > > >> >> > > >> >> T(1) U ... U T(n) > > >> > > > >> > That is completely sufficient as long as there is no upper bound, > > >> > but > > >> > n --> oo (potential infinity). > > >> > > >> Then I would like to have an explanation what > > >> > > >> T(1) U T(2) U ... > > >> > > >> shall mean in contrast to > > >> > > >> T(1) U T(2) U ... U T(n) n e N. > > > > > > There is no contrast but identity (for n --> oo). > > > > "M�ckenheim-Limes"? > > > There is a tree T which contains the root node at level 0 and if it > contains the (finite tree T(n) down to) level n then it contains the > (finite tree T(n+1) down to) level n+1. An n+1 level tree may contain an n level tree as a subtree, perhaps, but not as a subset, and unions are only defined for sets. In particular, no path in the level n tree is a path in the n+1 level tree. What WM is suggesting is not union but something else, for which he should find some other name. Perhaps a "conjunction" of trees would convey his intent. > The existence follows from the existence of the real numbers and all > their initial segments Real numbers do not have "initial segments". Does WM mean the initial segments of the binary representations of such numbers? > > The union of all levels = the union of all finite trees = tree T. If by this WM means the result is the complete infinite binary tree then, while its set of nodes is countable and its set of edges is also countable, its set of paths is not, as has been adequately demonstrated by Cantor.
From: imaginatorium on 23 Jan 2007 13:56 Andy Smith wrote: > G. Frege <nomail(a)invalid.?.invalid> writes > >On Tue, 23 Jan 2007 10:45:22 GMT, Andy Smith > ><Andy(a)phoenixsystems.co.uk> wrote: <snip> > >Huh? Where did you get that nonsense from. (Did you read one of > >Mückenheim's papers? :-? > > > No, who he? You've escaped noticing? All around you in this thread is the endless spewing of our German friend who, inexplicably, is said to _teach_ mathematics at some sort of college. He's just reponded and "confirmed" that your confusion is "correct". (But all in undefined verbiage.) > So Cantor's argument doesn't rely on locating an actually infinite > member of his list as some limit process n->oo, he just posits that an > actually infinite list can exist and is complete, and then shows it > can't be (or possibly alternatively that an actually infinite list can't > exist?) There are three "actually"s in the above statement. For each, contemplate carefully whether it makes the following word mean more or less... > OK, well I do see the argument better now, but if that was an argument > that I had suggested for the first time, you would be on me like wolves I've slightly lost what this refers to, but yes, you will get jumped on from time to time. All perfectly normal - typically someone just misread what you wrote. Brian Chandler http://imaginatorium.org
From: G. Frege on 23 Jan 2007 13:55
On Tue, 23 Jan 2007 13:48:45 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >> >> It is impossible that the "whole" set of even numbers has a cardinal number >> which is larger than every even number. >> > Why? > Because WM thinks so. F. P.S. You couldn't resist. Right? ;-) -- E-mail: info<at>simple-line<dot>de |