Cantor Confusion
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From: Virgil on 30 Sep 2006 14:44 In article <JguTg.1862$3E2.1091(a)tornado.rdc-kc.rr.com>, "Poker Joker" <Poker(a)wi.rr.com> wrote: > "Tonico" <Tonicopm(a)yahoo.com> wrote in message > news:1159605340.208078.275580(a)c28g2000cwb.googlegroups.com... > > > Iff you gave a complete list of the reals then the other part would NOT > > be able to construct a real that is not in it...kid, > > DAH! And any construction of such a number would be flawed. Only if some flaw could be proved, and PJ has proved nothing. > Perfectly grammatically correct. Just like the process that takes > all reals and produces on that wasn't in its input. The output LOOKS > good. No proof that if you feed it ALL the reals that it actually > produces as advertised. It can't. One does not "feed it all the reals" one feeds it all the lists of reals, and by a method independent of which list it is fed, it produces a number not listed by that list. In fact, by a suitable modification, it can produce one number not in the list for every real number, i.e., show a bijection between the set of all reals and, for any given list, the set of unlisted reals. So in this sense, for any fixed list, you can feed in all the reals and get for each real an unlisted real.
From: Arturo Magidin on 30 Sep 2006 15:03 In article <UeiTg.25580$QT.16960(a)tornado.rdc-kc.rr.com>, Poker Joker <Poker(a)wi.rr.com> wrote: > >"Arturo Magidin" <magidin(a)math.berkeley.edu> wrote in message >news:efj3bk$120f$1(a)agate.berkeley.edu... >> In article <N_YSg.1208$3E2.403(a)tornado.rdc-kc.rr.com>, >> Poker Joker <Poker(a)wi.rr.com> wrote: >>> >>>"Arturo Magidin" <magidin(a)math.berkeley.edu> wrote in message >>>news:efgfhd$261u$1(a)agate.berkeley.edu... >>>> In article <1159410937.013643.192240(a)h48g2000cwc.googlegroups.com>, >>>> <the_wign(a)yahoo.com> wrote: >>>>>Cantor's proof is one of the most popular topics on this NG. It >>>>>seems that people are confused or uncomfortable with it, so >>>>>I've tried to summarize it to the simplest terms: >>>>> >>>>>1. Assume there is a list containing all the reals. >>>>>2. Show that a real can be defined/constructed from that list. >>>>>3. Show why the real from step 2 is not on the list. >>>>>4. Conclude that the premise is wrong because of the contradiction. >>>> >>>> This is hardly the simplest terms. Much simpler is to do a ->direct<- >>>> proof instead of a proof by contradiction. >>>> >>>> 1. Take ANY list of real numbers. >>>> 2. Show that a real can be defined/constructed from that list. >>>> 3. Show that the real from step 2 is not on the list. >>>> 4. Conclude that no list can contain all reals. >>>> >>> >>>How can it be simpler if the list can be ANY list instead of a >>>particular one. >> >> Because a direct proof is simpler than a proof by contradiction. >> >>> ANY list opens up more possiblities than >>>a single list. >> >> Any list does not require you to assume that there is a "single list" >> which some some particular property. > >We all noticed you neglected this logic: Hardly. >if its true for ANY list, then it must be >true for a specific list. Indeed. >So if considering a single specific list >shows a flaw, then looking at ANY (ALL of them) list doesn't >help. And if my grandmother had wheels, then she'd be a bicycle. Since she doesn't, a discussion about just what kind of bicycle she might be in that case is a waste of time. Likewise, since no flaw has been exhibited by looking at any specific list (and "specific" in this case must mean explicit and specific, not a putative list with putative properties whose existence cannot be established a priori; otherwise, we might just say "take a list for which the argument does not work", which is of course nonsense), discussions about this are a waste of time. -- ====================================================================== "It's not denial. I'm just very selective about what I accept as reality." --- Calvin ("Calvin and Hobbes" by Bill Watterson) ====================================================================== Arturo Magidin magidin-at-member-ams-org
From: Tonico on 30 Sep 2006 15:20 Poker Joker wrote: .........nonsenses and rather weird attacks...... > So you think a process that takes all reals can produce one that isn't > in the set of all reals is possible? No, I don't...do you? And what process are you talking about? So far you've only mentioned the rather huge nonsensical mumbo-jumbo "if a list [sic] contains all the reals then trying to build a real that is not there...blah-blah"....And if the natural number 4 were a multiple of 3 then Moscow would be Rwanda's capital city: from a false statement ANYTHING can be deduced. What? That there can be a list [sic] containing all the reals? Prove it. I can prove, mathematically and not with ranting, that it can't be so > Wait, I get it. You want to use the consequences of having such > a process to prove that there is such a process. Now I understand > your logic. Well, it's refreshing to know you're understanding something at least and at last, even if it is something produced by your own imagination. > > IF you actually gave > > a complete "list" of the reals, uh?? By your """logic""" then, it is > > IMPOSIBLE to rebuke > > anything you say, because IF it is true then the rebuttal will have, OF > > COURSE!, to be flawed....great! > > It took you this long to figure that out? Of course, you can't stop > thinking that the conclusion stands on its own. So how could you > see the flaw? Hehe...oh well: just another crank. Wanna buy a bridge with a river under it? Tonio
From: Dik T. Winter on 30 Sep 2006 19:32 In article <LkiTg.25581$QT.14011(a)tornado.rdc-kc.rr.com> "Poker Joker" <Poker(a)wi.rr.com> writes: > "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote in message > news:J6CsBJ.Jys(a)cwi.nl... .... > > > Under the most general assumption, we can't count out that > > > R is f's image, so defining a real in terms of the image of > > > f *MIGHT* be self-referential, and it certainly is if the image > > > of f is R. > > > > What is the problem here? > > I assume you accept this proof that there are no complete lists > of reals: > > Let r be a real number between 0 and 1. Let r_n denote the nth digit > in r's decimal expansion. Let r_n = 5 if r_n = 4, otherwise let r_n = 4. > r isn't on any list of reals. Therefore there isn't a complete list of > reals. There is no r that satisfies that condition. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 30 Sep 2006 19:47
In article <vOjTg.25590$QT.1504(a)tornado.rdc-kc.rr.com> "Poker Joker" <Poker(a)wi.rr.com> writes: .... > For ANY x, there is procedure to construct a y, such that x = 1/y. How do you propose to prove that? > If everybody neglects the fact that the construction isn't valid > for x=0, then the proof is flawless. What proof? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |