A BLATENT FLAW in Cantor's diag proof
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From: herbzet on 12 Jun 2010 20:01 Pol Lux wrote: > On Jun 12, 8:07 am, "|-|ercules" <radgray...(a)yahoo.com> wrote: > > "Daryl McCullough" <stevendaryl3...(a)yahoo.com> wrote > > > > > > > > > > > > > |-|ercules says... > > > > >>I understand fully every point made against me. I also happen to see > > >>the error in your posts. > > > > > If that were true, then you could easily show it. Give the standard > > > proof of the uncountability of the reals, in a form that a mathematician > > > would agree: yes, that's a correct proof. It should take the form of > > > a sequence of statements such that each statement is either a definition, > > > an accepted mathematical fact, or follows from previous statements by > > > accepted rules of inference. > > > > > Then, what you need to do is to point out which step is incorrect. > > > If you really gave a proof, that means that either you find an axiom > > > (an accepted fact of mathematics) that you disagree with, or you find > > > an accepted rule of inference that you disagree with. > > > > > Then, you need to explain *why* you disagree with that axiom or rule > > > of inference. The most convincing explanation would be a proof (in > > > the sense that mathematicians would consider it a proof) that the > > > questionable axioms and/or rules of inference lead to a contradiction. > > > > > Short of that, you can show that they lead to a conclusion that is > > > contrary to other statements you feel ought to be true (even if they > > > are not provable). > > > > > You have not done either of these. You haven't shown that > > > standard mathematical axioms and rules of inference lead > > > to a contradiction. You have not proposed alternative axioms > > > that standard mathematics conflicts with. > > > > > The fact that you've done neither of these, and still claim to > > > have discovered a "blatant flaw" in Cantor's proof, shows that > > > you are mistaken. You don't understand Cantor's proof. > > > > > -- > > > Daryl McCullough > > > Ithaca, NY > > > > I accept your challenge! > > > > Cantor's diag proof - by Herc! > > > > Take any list of real expansions > > > > 123 > > 456 > > 789 > > > > Diag = 159 > > Anti-Diag = 260 > > > > VOILA - SUPERINFINITY! > > > > Herc > > It's wonderful to be able to see infinity so clearly in those 3 lines > of 3 numbers. Remarkable. That's nothing -- with a solid hit of mescaline you can see infinity in a grain of sand. -- hz
From: |-|ercules on 12 Jun 2010 20:19 "Daryl McCullough" <stevendaryl3016(a)yahoo.com> wrote ... > |-|ercules says... >> >>"Daryl McCullough" <stevendaryl3016(a)yahoo.com> wrote ... >>> |-|ercules says... >>> >>>>I accept your challenge! >>>> >>>>Cantor's diag proof - by Herc! >>>> >>>>Take any list of real expansions >>>> >>>>123 >>>>456 >>>>789 >>>> >>>>Diag = 159 >>>>Anti-Diag = 260 >>>> >>>>VOILA - SUPERINFINITY! >>> >>> Okay, so the first part of the challenge was to see >>> if you could give the standard proof of the uncountability >>> of the reals. You failed that part. >>> >>> The second part is to demonstrate what was wrong with >>> the proof in the first part. >>> >>> But you don't actually know how to do the first part. >>> So you were not telling the truth when you said that >>> you understood the standard proof. >>> >> >>Didn't you see my reproof, better than that one? > > You claimed to accept my challenge, and you failed. > That shows that you don't understand the subject. > You didn't even reply to my question 4 lines up so who cares about your shoddy arguments. OK Daryl, this is your last chance to engage the argument. "George Greene" <greeneg(a)email.unc.edu> wrote > On Jun 8, 4:29 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: >> YOU CAN'T FIND A NEW DIGIT SEQUENCE AT ANY POSITION ON THE COMPUTABLE REALS. > > OF COURSE you can't find it "at any position". > It is INFINITELY long and the differences occur at INFINITELY MANY > DIFFERENT positions! Is George right? Herc
From: Daryl McCullough on 13 Jun 2010 08:02 |-|ercules says... >OK Daryl, this is your last chance to engage the argument. Oh, my gosh! My last chance to weigh in on this topic! I hope I don't blow it! >"George Greene" <greeneg(a)email.unc.edu> wrote >> On Jun 8, 4:29 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: >>> YOU CAN'T FIND A NEW DIGIT SEQUENCE AT ANY POSITION ON THE >>> COMPUTABLE REALS. >> >> OF COURSE you can't find it "at any position". >> It is INFINITELY long and the differences occur at INFINITELY MANY >> DIFFERENT positions! > >Is George right? Yes, and you can't understand what he's saying because you don't understand the meaning of quantifiers. Let r_0, r_1, ..., be a list of reals. Let d be the anti-diagonal formed from this list. Then the facts are: d is unequal to r_0 d is unequal to r_1 d is unequal to r_2 In general, for each n, d is unequal to r_n Does that mean that there is a *finite* sequence of digits appearing in d that does not appear in r_0, r_1, etc.? No, it doesn't mean that! It could very well be the case that every finite sequence of digits appears somewhere on the list. So it could be that both of the following are true: 1. Given any n, there is a real r_m on the list such that d and r_m agree in the first n decimal places. 2. Given any real r_m on the list, there exists an n such that d and r_m do *not* agree in the first n decimal places. It is possible for *both* statements to be true simultaneously. The first sentence says "d does *not* have any new finite sequences of digits that are not already on the list". The second sentence says that "d is unequal to any real on the list". So d is not on the list, but it is *approximated* by elements on the list. For example, consider the list 3 3.1 3.14 3.141 3.1415 3.14159 etc. The real number pi is not on this list, because pi is not a terminating decimal, and each real on the list is a terminating decimal. On the other hand, pi doesn't have any *finite* digit sequence that isn't somewhere on the list. -- Daryl McCullough Ithaca, NY
From: Ross A. Finlayson on 14 Jun 2010 03:05 On Jun 13, 5:02 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > |-|ercules says... > > >OK Daryl, this is your last chance to engage the argument. > > Oh, my gosh! My last chance to weigh in on this topic! I hope I don't > blow it! > > >"George Greene" <gree...(a)email.unc.edu> wrote > >> On Jun 8, 4:29 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: > >>> YOU CAN'T FIND A NEW DIGIT SEQUENCE AT ANY POSITION ON THE > >>> COMPUTABLE REALS. > > >> OF COURSE you can't find it "at any position". > >> It is INFINITELY long and the differences occur at INFINITELY MANY > >> DIFFERENT positions! > > >Is George right? > > Yes, and you can't understand what he's saying because you don't > understand the meaning of quantifiers. > > Let r_0, r_1, ..., be a list of reals. Let d be the anti-diagonal > formed from this list. Then the facts are: > > d is unequal to r_0 > d is unequal to r_1 > d is unequal to r_2 > In general, for each n, d is unequal to r_n > > Does that mean that there is a *finite* sequence of digits > appearing in d that does not appear in r_0, r_1, etc.? > No, it doesn't mean that! It could very well be the case > that every finite sequence of digits appears somewhere on > the list. So it could be that both of the following are > true: > > 1. Given any n, there is a real r_m on the list such that > d and r_m agree in the first n decimal places. > > 2. Given any real r_m on the list, there exists an n > such that d and r_m do *not* agree in the first n > decimal places. > > It is possible for *both* statements to be true simultaneously. > The first sentence says "d does *not* have any new finite sequences > of digits that are not already on the list". The second sentence > says that "d is unequal to any real on the list". > > So d is not on the list, but it is *approximated* by elements > on the list. For example, consider the list > > 3 > 3.1 > 3.14 > 3.141 > 3.1415 > 3.14159 > etc. > > The real number pi is not on this list, because pi is not a terminating > decimal, and each real on the list is a terminating decimal. On the > other hand, pi doesn't have any *finite* digit sequence that isn't > somewhere on the list. > > -- > Daryl McCullough > Ithaca, NY Then I say this reasoning and the equivalency function work this way: when the list starts with each element of the list, their antidiagonal is 1, .111111..._2, that number is the end of the list, that's where it goes. The properties of the function have that they're symmetric about one half. (Here, after reading this and writing some more here, an actual idea might be to, with the antidiagonal of the equalency function, show algorithms for how various partitionings of the range where besides integer successorship there is also symmetry partitioning, that the resulting antidiagonals are of particular forms or have various limits, to forms antidiagonals beside the last list elements.) Now Herc here is casting about in some less than very coherent manner, that's not much direction, where I say within a framework of addressing mathematical constructions about the countability and uncountability of infinite sets, and between countability and uncountability, there is at least one particular function with the properties that a) the function has properties consistent with enough other functions that generally useful results are mathematically sound using this function and b) the function is between and onto from the natural integers, the counting numbers, to each of the elements of the continuum of the real numbers between zero and one, in order. There are a lot of series approximations for pi, each Cauchy is the same. There are a lot more definitions for the rationals than irrationals, for each one, that is. Basically I figure to write an article with the approximately five reasoning modes of a countability argument, where, as I have already written, each has for the equivalency function that it is special, thus of particular relevance to uncountability, and whether some sets are provably countable. Then, in this manner, it's a just alternative that works with the classical. As well then there is c) the function has enough interesting properties, for example that it is a natural probability distribution of the natural integers, that it then works with even more mathematics, where then that is just the beginning, of generally useful frameworks. It's not a blatant flaw, per se, blatant, yet, still it sees in working through each case that the resulting numerical assumptions affect the foundations, with a resolution where measure foundation is convenient of the equivalency function, that for me, I found it interesting mathematically and felt that it really helps generally the understanding of specific foundations, for example the regular foundations that are the standard in modern mathematics. There's still the uncountable in the regular, I go to nonstandard foundations to say ZF is inconsistent, to get back to having a regular foundation for measure theory. Furthermore, I explain how it can be shown it's for its particular properties the only function so. So personally when I heard that people did not see this as obvious and intuitive I found it serendipitous to think that I could explain it to them. Particularly getting through the ordinal induction schema, that really helps in just another typical mathematical evolution. Yet then of course I found that that is in the mathematics the most definite possible reduction to correctness that it wasn't enough to simply note the emperor's no clothes, had to make new ones to show they weren't. This went on for some time. So, about Cantor's infinite set theory, or rather about the part of Cantor's infinite set theory about uncountable sets and about the real numbers, a particular function is defined with properties that within each of various proof results has the natural integer continuum enough similar to the real number continuum that the real numbers of one line piece aren't uncountable, they are countable. (Excuse me, this isn't relevant to aus.tv.) Regards, Ross F.
From: Daryl McCullough on 14 Jun 2010 06:28
Ross A. Finlayson says... >Then I say this reasoning and the equivalency function work this way: >when the list starts with each element of the list, their antidiagonal >is 1, .111111..._2, that number is the end of the list, There is no end to the list, since there is no largest natural number. -- Daryl McCullough Ithaca, NY |