Cantor Confusion
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From: Franziska Neugebauer on 11 Jan 2007 18:11 mueckenh(a)rz.fh-augsburg.de wrote: > > Virgil schrieb: > >> In article <1168511880.370120.180940(a)p59g2000hsd.googlegroups.com>, >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> > Here is the formal proof: >> > >> > Theorem. The set of real numbers in [0, 1] is countable. >> >> Your proof was neither formal nor valid. >> >> Among other things you invalidly assume an infinite the union of >> finite sets must contain an infinite object, rather than merely >> containing infinitely many finite objects. > > An infinite union of finite sets is an infinite object. But that object does *not* have an infinite _member_. F. N. -- xyz
From: Virgil on 11 Jan 2007 18:17 In article <1168552370.844108.84970(a)p59g2000hsd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > > The "tallest man" is something that can change. L_D is a line. > > > > A line cannot change. The analogy is not valid. > > > > > > L_D is the name of a line, like a championship title. > > > > A championship title is not the name of a team. > > No. It is an attribute of a team. > > > "The Chicago Bulls" is the name of a team. > > "The NBA champions" is not the name of a team. > > A championship title refers to a team, but the > > team it refers to can change. The name of a > > team refers to a team, but the team it refers to > > cannot change. L_D is the name of a > > line. The line it refers to cannot change. > > Then use the name "largest line". Why not use "non-existent line"? It makes as much sense. > > Depends what you mean by "the whole set". > > If you mean every element of the diagonal > > that can be shown to exist, then "the line" does not exist. > > If you mean "all elements > > of the diagonal that have been shown to exist", > > then "the whole set" is something that can > > change and "the line" must be something > > that can change. > > So it is. > > > > In either case "the line" is not L_D. > > In both cases there is a largest line, namely the largest line which > can be shown to exist or the largest line which has been shown to > exist. The first one is obviously a contradiction, it refers to actual > infinity. Hence, our only choice is the second. Speak only for yourself. For many of us who choose to work in ZFC or NBG or equivalent systems, the second is as contradictory as the first, since showing the existence of any line proves the existence of a longer one. > > > Let L_D go from 1 to oo. > > > > This statement is meaningless. > > L_D is a line. Lines do not "go from 1 to oo". > > Not individual lines, but largest lines. Such self-contradictory lines go anywhere they want.
From: Virgil on 11 Jan 2007 18:19 In article <1168552885.290846.124430(a)p59g2000hsd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > You are unable to explain a mathematical meaning. I thought you were a > > teacher. > > Yes, but I am not a God. You claim the authority of one by dictating what others should be allowed to think. > > It is so easy: Potential infinite sets are finite. But their maximum is > not fixed. So that in WM's mathematics constants are variable.
From: Franziska Neugebauer on 11 Jan 2007 18:28 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> mueckenh(a)rz.fh-augsburg.de wrote: >> > Franziska Neugebauer schrieb: [...] >> > The set of paths is countable. >> >> 1. The set of all rooted _finite_ paths in an infinite tree is >> countable. The union of all finite paths is exactly this set of >> all rooted finite paths. > > Yes, of infinitely many paths, each of which has infinitely many > nodes. _finitely_ many nodes. Your trees to be united do only have finitely many nodes. There are only finitely deep trees. Hence the union does not contain any infinite tree and hence not any infinite path. Recall: Do get it out you must have put it in. [...] >> 2. To "unary represent" every real in [0, 1] > > You cannot "unary represent" every real in [0, 1]. It's rather unimportant whether I "_can_" do so. But for the sake of argument let us assume, you "could" *not* represent every real in [0, 1] in terms of representations. Then your proof is meaningless already before writing it down. F. N. -- xyz
From: Franziska Neugebauer on 11 Jan 2007 18:35
mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> > There are mathematicians greater than you and any living set >> > terrorists which are convinced of the opposite. >> >> I don't keep company with terrorists. > > Pardon, a typing error. I meant set-theorists, of course. Excuse me. You're welcome. >> >> From this follows: "A potentially infinite quantity (set or not) >> >> is always finite" is a meaningless sentence. >> > >> > Everything you don't understand seems meaningless to you. >> >> You are unable to explain a mathematical meaning. I thought you were >> a teacher. > > Yes, but I am not a God. Thanks for the clarification. I will notify EB. >>I suppose the deeply buried truths of ancient >> mathematicians will remain inaccessible to me. If there are any at >> all. > > It is so easy: Potential infinite sets are finite. But their maximum > is not fixed. "mutable sets" is a notifiable disease. F. N. -- xyz |