Cantor Confusion
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From: Virgil on 31 Oct 2006 19:22 In article <1162330445.200359.82310(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > > > Here it means, the result of my proof is valid for all possible > > > theories which allow to define the infinite binary tree. > > > > Sorry. I still don't know what you mean. ZFC is enough to define an > > infinite binary tree. Are you saying that your proof can be done in ZFC > > or are you not saying this? If not, then what are you saying? > > If ZFC is enough to define an infinite binary tree, then it should be > enough to see that there are more edges than paths. WM claims that proving 1 + 1 = 2 will allow us to prove 1 = 2? There is nothing in the construction of an infinite binary tree, in any consistent system, that allows one to prove that there are more edges than paths. > Where is a gap? That WM does not see the gap is testament to his blindness. > > > > > Either you have a proof that can be given in ZFC or you don't. Which is > > > > it? This shouldn't be a difficult question to answer. > > > > > > But it is not an interesting question. > > > > Perhaps not interesting to you, but please answer it anyway. > > I know of a really good logician who told me that he is unable to > translate my proof into ZFC. I don't know what is missing. Neither does > he. But, as I said, I will not learn to calculate horoscopes in order > to judge that astrology is nonsense. Horoscopes make as much sense as WM makes. Each edge terminates in a node. Each node trivially matches a unique finite binary string. Each path trivially determines an endless binary string. So WM claims that the set of finite binary strings surjects to the set of endless binary strings. But WM cannot display any such surjection, as none exists. > > Please give one letter which requires an actually infinite number omega > of columns. > If you can't, please stop the nonsense talk about an infinite number of > finite numbers. Please give one letter which cannot occur in an infinite number of columns by beong the nth letter in the nth row and nth column. If you can't, please stop the nonsense talk about there not beding an infinite number of finite numbers. > > > > If the list consists of finite sequences, then the diaogonal is a > > > finite sequence too. Because it cannot be broader than the list As the list has no limit on "broadness" when the nth member has at least n characters, WM is off his rocker again. > > > In ZFC or in some other system? > > Everywhere. As the list has no limit on "broadness" when the nth member has at least n characters, WM is off his rocker again, everywhere.
From: David Marcus on 31 Oct 2006 19:23 mueckenh(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > > > All entries of the list have a finite number of letters. > > > > Correct. And given any integer, we can take a set of lines such that > > the number of letters in this set is greater than the integer. > > Correct. And given any set of lines we can find a finite number which > is larger. Otherwise, at least one of the lines would contain a number > of letters which was larger than any natural number, i.e. which was > omega. > > > > > An infinite > > > sequence is larger than any finite sequence. The diagonal of a list > > > cannot have more letters than the lines. > > > > > > > Correct. The number of letters in the lines is greater than any > > integer. > > Incorrect. Greater than any integer is no integer but only omega (an > its successors). > > > So the greatest number of letters the diagonal can have is > > greater than any integer. > > Incorrect. See above. > > > > So the diagonal has infinite length (call this potentially infinite > > length if > > you get your kicks by saying potentially). > > The diagonal may have potentially infinite length, but that is less > than actual infinity, i.e., omega which is the first number larger than > any natural number. > > Die Anzahl einer unendlichen Menge [ist] eine durch das Gesetz der > Z=C3=A4hlung mitbestimmte unendliche ganze Zahl. (G. Cantor, Collected > Works p. 174) > .=2E. kann also omega sowohl als eine gerade, wie als eine ungerade Zahl > aufgefa=C3=9Ft werden. (G. Cantor, Collected Works p. 178) > Es ist sogar erlaubt, sich die neugeschaffene Zahl omega als Grenze zu > denken, welcher die Zahlen nu zustreben, wenn darunter nichts anderes > verstanden wird, als da=C3=9F=EF=80=A0omega die erste ganze Zahl sein soll, > welche auf alle Zahlen folgt, d. h. gr=C3=B6=C3=9Fer zu nennen ist als jede > der Zahlen n. (G. Cantor, Collected Works p. 195) Don't you think that you should label all your posts as "NON-STANDARD MATHEMATICS"? That would prevent people from accidentally reading them and being confused because you've changed the meaning of all the standard terminology to suit yourself. If you want to redefine all the words in mathematics (e.g., "number", "finite", "infinite", "sequence", "contradiction", "countable", "uncountable"), then feel free, but don't pretend that you aren't. -- David Marcus
From: Virgil on 31 Oct 2006 19:28 In article <1162330574.086231.91110(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <c5a1b$4545ba52$82a1e228$12545(a)news1.tudelft.nl>, > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > > > > > > Are you still doing physics, water, and a finite number of molecules? > > > > Let us know when you switch to mathematics. > > > > > > I'm DOING mathematics. Mathematics is NOT independent of Physics. > > > > That dependency is a cross only physicists bear. > > For non-physicists, mathematics is quite independent of physics. > > No. They only don't recognize the dependence. How can one recognize what does not exist? > >> Every member of a list of only finite sequences is a finite sequence. > > > But that does not limit the number of such sequences to being finite. > > You say so. But then the number of such sequences is larger than the > columns. However, it does not play any role. The diagonal has not more > elements than the sequences. WM blinds himself with faulty arguments. When the number of "columns" in the nth listed row is always greater than n, then the diagonal has more that any finite number of elements. > > > >> Every diagonal of such a list is a finite sequence. > > > Wrong! > > > Define the infinite sequence of finite strings in which the nth string > > consists of n 1's, and define the infinite diagonal so that in its nth > > position it has a "2" for each of infinitely many n's. > > It is ridiculous how the brain is washed by study of "logic". Did you > ever hear of a finite set which contains an infinite set? Did WM ever hear of an infinite set which did not contain a finite set? WM again washes his own brain with thought killing arguments.
From: Dik T. Winter on 31 Oct 2006 19:38 In article <1162317871.630561.228540(a)h48g2000cwc.googlegroups.com> "MoeBlee" <jazzmobe(a)hotmail.com> writes: > mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > What do you mean an axiom is "false"? > > > > An axiom leading to a contradiction when added to a set of axiom which > > do not so, is false. > > Under that definition of 'false', EVERY non-logical sentence is false. Including every axiom. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: David Marcus on 31 Oct 2006 19:40
mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > > Here it means, the result of my proof is valid for all possible > > > theories which allow to define the infinite binary tree. > > > > Sorry. I still don't know what you mean. ZFC is enough to define an > > infinite binary tree. Are you saying that your proof can be done in ZFC > > or are you not saying this? If not, then what are you saying? > > If ZFC is enough to define an infinite binary tree, then it should be > enough to see that there are more edges than paths. Where is a gap? is > ZFC not able to sum the geometric series? Or is i not possible to > consider fractions of elements? As lots of people have told you, your "proof" is either incomprehensible or wrong. You use words without defining them. When people assume a plausible meaning for the words (some standard meaning), the resulting argument is clearly incorrect. Therefore, either we are not correctly guessing your meaning or you are in error. Either way, if you claim a proof, then it is up to you to explain it. This is the way mathematicians have been doing things for thousands of years. In particular, in your argument, you use the word "relation". However, what you say your "relation" is does not qualify as a relation under the standard meaning of the word. It is up to you to either define a proper relation or explain what your non-relation is and why it is relevant. > > > > Either you have a proof that can be given in ZFC or you don't. Which is > > > > it? This shouldn't be a difficult question to answer. > > > > > > But it is not an interesting question. > > > > Perhaps not interesting to you, but please answer it anyway. > > I know of a really good logician who told me that he is unable to > translate my proof into ZFC. Hardly surprising. If he is truly a good logician, then he didn't want to tell you that you were talking gibberish. > I don't know what is missing. We've noticed. Of course, if you weren't so certain that you were right (and everyone else is wrong), you might learn what is missing by listening to what people say. > Neither does he. Again, not surprising he would say that. > But, as I said, I will not learn to calculate horoscopes in order > to judge that astrology is nonsense. Fine. Then stop claiming that you can prove ZFC is inconsistent. If you don't like ZFC, that's your concern. But, that is completely different from saying that you have a proof within ZFC of a contradiction. > > OK, let's use our logic (or standard logic or ZFC). The list has > > infinitely many lines and columns, so the number of diagonal elements is > > infinite. Your statement that the number of diagonal elements is finite > > is wrong. You've jumped from "each entry is finite" to "the number of > > columns is finite". > > Please give one letter which requires an actually infinite number omega > of columns. > If you can't, please stop the nonsense talk about an infinite number of > finite numbers. You seem to have skipped right over my statement that I was using standard logic (or ZFC). In standard logic (or ZFC), "the number of columns is finite" is not the same as "each entry in the list has a finite number of columns". It seems simply bizarre that you would mix these up. > > > If the list consists of finite sequences, then the diaogonal is a > > > finite sequence too. Because it cannot be broader than the list > > > In ZFC or in some other system? > > Everywhere. How would you know since you admitted above that your proofs don't work in ZFC? Why make claims that you then immediately contradict? -- David Marcus |