THE HOLY GRAIL ~ CANTOR DISPROOF
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From: William Hughes on 24 May 2010 20:39 On May 24, 9:04 pm, Herc7 <ozd...(a)australia.edu> wrote: > On May 25, 9:38 am, Transfer Principle <lwal...(a)lausd.net> wrote: > > > > > On May 24, 1:40 pm, Herc7 <ozd...(a)australia.edu> wrote: > > > > On May 25, 2:46 am, Transfer Principle <lwal...(a)lausd.net> wrote: > > > > Case 1. Herc/Cooper is trying to propose a new > > > > theory in which only computable reals count as > > > > reals in the proposed theory. > > > > Case 2. Herc/Cooper knows that standard theory > > > > proves the existence of uncountably many reals, > > > > but doesn't like this fact. > > > > Case 3. Herc/Cooper believes that the standard > > > > theory proves that there are only countably many > > > > reals, and that he has the proof. In this case, > > > > we might say that Herc/Cooper is wrong. > > > > In another thread, some posters have suggested > > > > that I add a fourth case. In Case 4, we say > > > > that Herc/Cooper is "not even wrong," because > > > > his claim is so outrageous. > > > I have made no claim in this post that there are only countable reals, > > > I merely show that Cantor's proof by diagonalization > > > is inconsistent with probability theory. > > > OK then, we can safely eliminate Case 1. > > > > Try answering the question I put forth to you 3 days ago. > > > If the computable reals can be shuffled to fit any randomly > > > generated diagonal > > > I have yet to see a valid proof in ZFC that this holds. > > > To me, this _conjecture_ is a bit strange. After > > thinking about it for a while, I'm starting to > > wonder whether it's even plausible. > > > Notice that Herc/Cooper claims to have proved it, > > and even once gave a link to a website which > > demonstrates the conjecture by providing a random > > list and diagonal, then reorders the list such that > > the diagonal matches the given real. But this was > > flawed for two reasons: > > > 1. The list and diagonal real were both _finite_. > > 2. I was able to stump the computer by choosing my > > own real for the diagonal, and the computer wasn't > > able to find a permutation of the list for it. > > > Of course, Herc/Cooper was discussing infinite > > lists, and so a counterexample for a finite list > > doesn't disprove the infinite case. > > > But, after thinking about it for a while, I believe > > to have found an infinite counterexample after all. > > > Let's say we want the diagonal to be: > > > D = .020202... > > > This is the rational number 1/4 (in ternary). So > > according to Herc/Cooper, there exists a permutation > > of the list of computable reals such that this number > > lies on the diagonal. This list contains every > > computable real, so in particular, it contains: > > > R = .111111... > > > This is the rational number 1/2, which, being > > computable, must be on the list. But where is it? It > > can't be the first number, since the first digits of > > D and R don't match. It can't be the second number, > > since the second digits of D and R don't match. It > > can't be the third number, since the third digits of > > D and R don't match. And so on. > > > Thus the computable reals can't be shuffled to fit the > > diagonal 1/4. Therefore, the conjecture is false. QED > > > And so we can snip the rest of Herc's question, since > > its premise is false. The computable reals can't be > > shuffled to fit any diagonal. > > You are selecting specific diagonals based on the list. > > The probability of fitting a random diagonal to the list of computable > numbers resulting in a different set of numbers is 0. > > Herc There are however an infinite number of counterexamples. Interestingly this set is uncountable. - William Hughes
From: Aatu Koskensilta on 25 May 2010 13:11 Transfer Principle <lwalke3(a)lausd.net> writes: > On May 24, 1:40�pm, Herc7 <ozd...(a)australia.edu> wrote: > >> Try answering the question I put forth to you 3 days ago. >> If the computable reals can be shuffled to fit any randomly >> generated diagonal > > I have yet to see a valid proof in ZFC that this holds. There is none. As you note diagonalising a list of the computable reals always results in an irrational, simply by observing all rationals are computable. > Notice that Herc/Cooper claims to have proved it, Notice that Herc/Cooper claims to be both Genesis Adam and Truman, reports he hears voices, professes to have psychic abilities, etc. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Don Stockbauer on 25 May 2010 14:45 On May 25, 12:11 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Transfer Principle <lwal...(a)lausd.net> writes: > > On May 24, 1:40 pm, Herc7 <ozd...(a)australia.edu> wrote: > > >> Try answering the question I put forth to you 3 days ago. > >> If the computable reals can be shuffled to fit any randomly > >> generated diagonal > > > I have yet to see a valid proof in ZFC that this holds. > > There is none. As you note diagonalising a list of the computable reals > always results in an irrational, simply by observing all rationals are > computable. > > > Notice that Herc/Cooper claims to have proved it, > > Notice that Herc/Cooper claims to be both Genesis Adam and Truman, > reports he hears voices, professes to have psychic abilities, etc. Now, now, now. Let's not fall victim to the "Argument Against the Man" (er, Person, nowadays).
From: Aatu Koskensilta on 25 May 2010 15:21 Don Stockbauer <donstockbauer(a)hotmail.com> writes: > Now, now, now. Let's not fall victim to the "Argument Against the > Man" (er, Person, nowadays). In this instance the information that Herc appears to be clinically insane is salient. Recall the nature of lwalker's valiant quest. He wants to vindicate people who get called (mathematical) cranks by other people by finding theories in which their unorthodox claims are provable. This sort of vindication obviously can work only if these theories have some actual connection to the thinking of the getting-called-a-crank sort of person in question. And as lwalke admits, it's possible these people are, in fact, just wrong about standard stuff, sometimes they simply don't like the mathematical facts, sometimes they are utter nutters, and so on, in which case the search for an appropriate vindicating theory is futile or irrelevant. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Virgil on 25 May 2010 20:27
In article <87vdabu6r4.fsf(a)dialatheia.truth.invalid>, Aatu Koskensilta <aatu.koskensilta(a)uta.fi> wrote: > Don Stockbauer <donstockbauer(a)hotmail.com> writes: > > > Now, now, now. Let's not fall victim to the "Argument Against the > > Man" (er, Person, nowadays). > > In this instance the information that Herc appears to be clinically > insane is salient. Recall the nature of lwalker's valiant quest. He > wants to vindicate people who get called (mathematical) cranks by other > people by finding theories in which their unorthodox claims are > provable. This sort of vindication obviously can work only if these > theories have some actual connection to the thinking of the > getting-called-a-crank sort of person in question. And as lwalke admits, > it's possible these people are, in fact, just wrong about standard > stuff, sometimes they simply don't like the mathematical facts, > sometimes they are utter nutters, and so on, in which case the search > for an appropriate vindicating theory is futile or irrelevant. I suppose that even a total and actual nutter could be accidentally right once in a long, long while. Though, of course, one would be foolish to count on it. |