THE CANTOR ARGUMENT SO FAR
in [Logic]
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From: Transfer Principle on 21 Jun 2010 23:56 On Jun 21, 8:49 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Jun 20, 8:14 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: > > "Sylvia Else" <syl...(a)not.here.invalid> wrote > > > Well, it's true that I rather assumed, without proof, that any number > > > can be created in the diagonal. But I don't think anything relevant > > > turns on that. > > You admit your error to him but not me? > But then again, Else made the same error that Herc made By which I mean the mistake that Herc made _years_ago_ when he came up with this idea of swapping to create a diagonal in the first place. Now, of course, he concedes that 0.222... can't appear in the diagonal if 0.111... is in the list, but only after either I (about a month ago) or some other poster showed him the error.
From: Graham Cooper on 22 Jun 2010 04:06 On Jun 22, 1:56 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Jun 21, 8:49 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > > On Jun 20, 8:14 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: > > > "Sylvia Else" <syl...(a)not.here.invalid> wrote > > > > Well, it's true that I rather assumed, without proof, that any number > > > > can be created in the diagonal. But I don't think anything relevant > > > > turns on that. > > > You admit your error to him but not me? > > But then again, Else made the same error that Herc made > > By which I mean the mistake that Herc made _years_ago_ > when he came up with this idea of swapping to create a > diagonal in the first place. > > Now, of course, he concedes that 0.222... can't appear > in the diagonal if 0.111... is in the list, but only > after either I (about a month ago) or some other poster > showed him the error. You are posting pure fantasy. I was well aware of rigged diagonals defeating my algorithm years ago that's why I claimed random diagonals will fit. You may have mentioned 0.11.. In some other argument but if you would provide a cite every time you say "herc was wrong" you would realize nobody has ever corrected me. Not even my bad spelling! Even the captcha is EURNOS. NO ERROR! Herc
From: James Burns on 22 Jun 2010 11:40 George Greene wrote: > On Jun 19, 6:21 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: > >>------------------------SCI.MATH----------------------------- >> >>Take any list of reals >> >>123 >>456 >>789 >> >>Diag = 159 >>AntiDiag = 260 > > Damn, you don't even know how to anti-diagonalize. > In an even base, the complement is 1 less than the base, > MINUS the number. > The "anti"- of d in decimal is 9-d. > Usually we do this in binary, where it is 1-b. > So the ACTUAL anti-diag here is > 840. No, GEORGE, you are pathetically wrong here. There IS NO "the" anti-diag. In order to be useful to the diagonal argument, an anti-diagonal needs its k-th digit to prove that the anti-diag is not equal to the k-th listed real. That is ALL it needs to do. An anti-diagonal cannot have the k-th digit as the k-th real and it cannot have a 0 or 9 as any digit, in order to avoid confusion over reals with multiple representations, like 0.1999... and 0.2000... . (I'm not sure about this last rule as a requirement -- there may be other ways to accomplish the same ends.) Therefore, any real with 2,3,4,5,6,7,8 as a first digit, 1,2,3,4,6,7,8 as a second digit, and 1,2,3,4,5,6,7,8 as a third digit is "the" antidiagonal for the three-real list under discussion. It would be rhetorically USEFUL to have a simple algorithm that would select one of the permissible digits for each place of the missing real, but one could make the digit selection arbitrarily complicated and the logic of the proof would still work. There is a point to this post, beyond sticking a pin into you, GEORGE, and watching you go pop. Many anti-Cantorians (not including Herc, in this case) try to get around the diagonal argument by squeezing "just one more" real into the list, as it is done in Hilbert's Hotel. To these people, I think it is useful to point out that there is no single "just one more" diagonal for their argument to work on. Indeed, there are just as many anti-diagonals (easily provable to be left off the given list) as there are reals, both listed and unlisted. (One can map the reals in base seven to the anti-diagonals in base ten very easily.) The anti-Cantorians think "We are almost done: we've listed every real except this anti-diagonal." However, they should more correctly be thinking that they have hardly begun: after listing as many reals as we can, in as clever a fashion as we can (perhaps by listing all the computable reals, followed by squeezing "just one more" a countable number of times), there are still just as many reals left unlisted as when we started. Jim Burns
From: Mike Terry on 22 Jun 2010 13:55 "Transfer Principle" <lwalke3(a)lausd.net> wrote in message news:11547e1f-04e5-4f87-8dcd-5b6208c23251(a)v29g2000prb.googlegroups.com... > On Jun 21, 5:59 pm, "Mike Terry" > <news.dead.person.sto...(a)darjeeling.plus.com> wrote: > > "Tim Little" <t...(a)little-possums.net> wrote in message > > > More generally still, it is not easy to define any reasonable meaning > > > for an infinite composition of non-disjoint permutations. > > Aaaaargh, it's all starting to come back! This is how I started off with > > Herc a few years ago, and Herc believed that IF he could permute the list in > > order to achieve ANY diagonal of his choosing, then he could engineer it so > > that the corresponding antidiagonal was in the list, although by Cantor's > > argument it is not. Well, this much is correct, but obviously it won't be > > possible to permute the list in this way... > > But then Herc went on to invent a succession of ever more complex schemes > > involving building up these "permutations" by successively > > swapping/shuffling rows of a carefully constructed starting list. Actually > > the whole idea was quite good fun to think through by comparison with > > today's Herc issues, and in the end the problem boiled down to that, say, > > row 1 would initially be swapped to row 20, and then in step 20 it would be > > swapped to say row 1000, and in step 1000...etc. > > Well, in the end row 1 would end up being shuffled right out of the list > > altogether! > > Ah yes, a bit similar to Hilbert's Hotel. > > I was wondering whether something like this could happen, but > Little's comment about 0.111... = 1/9 (or 1/2 in the original > ternary that Cooper is using) means that it is futile to > put 1/2 on the diagonal before worrying about this Hilbert's > Hotel paradox at all. > > But of course, I assume that Herc proposed some sort of > _algorithm_ to get the real r on the diagonal. It most likely > went something like this -- if the dth digit of the dth real > doesn't match the dth digit of r, then we swap the dth real > with the nth real, where n is the first natural greater than > d such that the dth digit of the nth real is the correct dth > digit of r. Since the list contains all the computable reals, > and there are infinitely many computable reals with the > correct digit in the dth place, and there have been only > finitely many reals before reaching the dth real, there's > guaranteed to be a computable real remaining in the list with > the correct digit in the dth place. Exactly. I believe he was working with an initial list that was crafted specifically to make it obvious that there would be suitable "swap line" later in the list when needed. So we end up with a countably infinite succession of swap operations, which Herc was assuming would still be just a permutation of the original list. > > Thus r is guaranteed to appear on the diagonal. But if r were > say, any member of the Cantor set (so that it contains only > zeros and twos in its ternary expansion), then 1/2 = 0.111... > will be swapped, and swapped again, indeed producing the > Hilbert's Hotel effect as described by Terry. > > Still, will this happen with _almost_every_ real that we try > to put on the diagonal? I don't know - there are uncountably many numbers not in the original list, so as has been said, there will be a lot of flexibility in the diagonals that can be produced this way. The point is really that it doesn't matter when it comes to arguing about Cantor's proof. (As a question in its own right, interesting maybe..)
From: George Greene on 22 Jun 2010 19:50
On Jun 20, 2:02 am, "|-|ercules" <radgray...(a)yahoo.com> wrote: > Hypothesis: a real number contains a finite sequence that is not computable. > > Contradiction Yes, it is true that THAT is a contradiction, since EVERY finite sequence is computable (the list of all of them is computable as well). > Therefore: all digits of every real are contained > in the list of computable reals. You HAVE NOT DEFINED *contained*, DUMBASS. More to the point, almost any definition OF "contained" THAT YOU COULD come up with would have the property that "all" (a word that should just be banned from discussions of this type; people should be required to say "every" or "any", INSTEAD) "digits of every real are contained in the list of" FINITE digit- strings. IF you would JUST SAY THAT, INSTEAD, THEN we could get somewhere. But the fact that you keep invoking a list that IS ITSELF UNcomputable (the list of all computable reals) just proves you're hopeless. |