Cantor Confusion
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From: Newberry on 18 Dec 2006 15:46 Virgil wrote: > In article <1166451538.861033.303300(a)48g2000cwx.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Newberry schrieb: > > > > > Sorry that I joined a bit late. > > Doesn't matter. > > > > >Are you saying that (in an infinite > > > binary tree) the set of paths is uncountable but the set of edges is > > > countable? > > > > The set of edges is obviously countable by, e.g., > > > > 1, 2, > > 3,4,5,6, > > 7,... > > > > As no path can separate from another one without the existence of two > > more edges, the number of edges is an upper bound for the number of > > paths. > > > That is only the case in finite trees. > It fails miserably in infinite binary trees in which no path has a last > node or last edge. Do you agree that the number of edges is the upper bound of the number of paths but disagree that the edges are countable? I guess if the number of edges > number of paths and you accept the diagonal argument then it follows that the edges are uncountable. I do not know if there is any way to prove directly that the edges are countable.
From: Franziska Neugebauer on 18 Dec 2006 15:46 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > >> >> > > > All elements that can be shown to exist in the diagonal can >> >> > > > be >> >> > > > shown to exist in one single line. [(P1)] >> >> This proposition P1 has _not_ yet been proved (shown). > > This proposition is the definition of the diagonal. 1. What is the definiendum? 2. What is the definiens? 3. The usual definition of the diagonal is (d_i) = (a_ii) i e omega. P1 is a proposition and not a definition. Mathematically P1 says (in my view) Em e omega Ak e omega (a_mk = a_kk) which is wrong since Am e omega Ak e omega(k > m & a_mk != a_kk), and it is also unproven. a_ij denotes the matrix of figures. >> > You misinterpret L_D. L_D is that line which contains all numbers >> > contained in the diagonal. >> >> The Diagonal is unbounded thus any _assumed_ L_D is not bounded, too. > > Correct is: > If the diagonal is unbounded then any _assumed_ L_D is not bounded, > too. 4. So you agree that unbounded(digonal) -> unbounded L_D? >>From the finiteness of any L_D we obtain: The diagonal is not > unbounded. > >> Hence L_D cannot be a line of the list (meaning: cannot be _in_ the >> list) for any line in the list is bounded (proof by induction). >> >> Hence contradiction to P1. P1 must be dropped. >> >> > If your L_D does not contain them, then you have the wrong L_D. >> >> Upto now we have no line L_D at all since P1 has not been proved. >> > There is nothing to prove in a definition. 5. P1 is not a definition. If you do not agree please answer the questions 1. and 2. > I remember you said a definition cannot be wrong. 6. I said that definitions are linguistic agreements (abbreviations) and that definitions _carry_ no truth value (are neither wrong nor right). 7. If one says that "a definition is wrong" it usually means that a "wrong" definiens is attached to the definiendum, or vice versa. > So it cannot be proved. 8. P1 is not a definition. It states: Em e omega Ak e omega (a_mk = a_kk) ,----[ <45863e06$0$97237$892e7fe2(a)authen.yellow.readfreenews.net> ] | The Diagonal is unbounded thus any assumed L_D is not bounded, too. | Hence L_D cannot be a line of the list (meaning: cannot be in the | list) for any line in the list is bounded (proof by induction). | | Hence contradiction to P1. P1 must be dropped `----. F. N. -- xyz
From: Virgil on 18 Dec 2006 15:47 In article <1166455447.296349.311760(a)73g2000cwn.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > Franziska Neugebauer schrieb: > > > > > > > > > > >> > > > All elements that can be shown to exist in the diagonal can be > > > > >> > > > shown to exist in one single line. [(P1)] > > > > > > > > This proposition P1 has _not_ yet been proved (shown). > > > > > > This proposition is the definition of the diagonal. > > > > > > > > You misinterpret L_D. L_D is that line which contains all numbers > > > > > contained in the diagonal. > > > > > > > > The Diagonal is unbounded thus any _assumed_ L_D is not bounded, too. > > > > > > Correct is: > > > If the diagonal is unbounded then any _assumed_ L_D is not bounded, > > > too. > > > > > > L_D is a line and is therefore bounded. > > > > Contradiction. > > Hence, the _assumed_ L_D does not exist. > > > Assumed is the diagonal. Premise: "If the diagonal is unbounded." > L_D is a line and is therefore bounded. > > Regards, WM The antecedent being false (if the diagonal is unbounded, no L_D line can exist), any consequent follows.
From: Virgil on 18 Dec 2006 15:52 In article <1166455723.089710.246540(a)f1g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Newberry schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > As no path can separate from another one without the existence of two > > > more edges, the number of edges is an upper bound for the number of > > > paths. > > > > If it is so simple why do you still need the proof by inheritance of > > shares? > > I don't need it. But there are others who do not see the forest because > of too many trees blocking the view. WM does need it to blind himself to the necessities of the situation. First WM denies infiniteness even in ZFC and NBG. Then WM discusses infinite binary trees, in which he divides an edge into countably many equal sized pieces. Oh, what a tangled web WM weaves,...
From: cbrown on 18 Dec 2006 15:58
mueckenh(a)rz.fh-augsburg.de wrote: > cbrown(a)cbrownsystems.com schrieb: > > > > > > Your edges are not > > > > being broken into 2 "shares" each; they are being divided into an > > > > infinite number of shares, and that is very different from 2 shares! > > > > > > For this sake we have mathematics, I mean really correct mathematics, > > > not based on dubious infinite sets, which states that 1 + 1/2 + 1/4 + > > > ... = 2. > > > > > > > "Really correct mathematics" doesn't simply /state/ that "1 + 1/2 + 1/4 > > + ... = 2". Mathematics /defines/ what the series of symbols "1 + 1/2 + > > 1/4 + ... = 2" means, and then /proves/ that it is true from these > > definitions. > > > That is the nonsense en vogue today. So you consider it nonsense that mathematics should consist of clear definitions and proofs of theorems. Instead you think mathematics should consist of a series of vague declarations, that may mean different things to different people, that one then claims supports a conclusion because one should feel it is "true in one's heart". In that case, it really is pointless to continue our discussion. Lord have mercy on your students. Cheers - Chas |