Cantor Confusion
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From: MoeBlee on 2 Oct 2006 20:32 Poker Joker wrote: > I asked a question. According to you, the question is incorrect. That's > cute. You also made a series of statements: You 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. ANY list opens up more possiblities than > a single list. Also, if its true for ANY list, then it must be > true for a specific list. So if considering a single specific list > shows a flaw, then looking at ANY (ALL of them) list doesn't > help. There is one sentence there that is a question, followed by three sentences that are statements. MoeBlee
From: Virgil on 2 Oct 2006 22:47 In article <PFhUg.27934$8_5.11593(a)tornado.rdc-kc.rr.com>, "Poker Joker" <Poker(a)wi.rr.com> wrote: > "Virgil" <virgil(a)comcast.net> wrote in message > news:virgil-343665.12274830092006(a)comcast.dca.giganews.com... > > In article <NNtTg.1856$3E2.504(a)tornado.rdc-kc.rr.com>, > > "Poker Joker" <Poker(a)wi.rr.com> wrote: > > > >> "Virgil" <virgil(a)comcast.net> wrote in message > >> news:virgil-988F09.00544830092006(a)comcast.dca.giganews.com... > >> > >> > It's not what Joker doesn't know that hurts him, its what he knows that > >> > ain't so. > >> > >> Virgil is jealous because he doesn't know anything other than to > >> get flame-wars going that allow him to use his talent: acting like > >> a three-year-old. > > > > I might be jealous of someone with some mathematical talent, but not of > > someone whose only visible talent is personal attacks. > > So you are your worst fan. PJ gets things as wrong as possible, as usual.
From: Virgil on 2 Oct 2006 22:49 In article <CHhUg.27967$8_5.24503(a)tornado.rdc-kc.rr.com>, "Poker Joker" <Poker(a)wi.rr.com> wrote: > "Virgil" <virgil(a)comcast.net> wrote in message > news:virgil-665CA8.12340030092006(a)comcast.dca.giganews.com... > > In article <Y3uTg.1861$3E2.1791(a)tornado.rdc-kc.rr.com>, > > "Poker Joker" <Poker(a)wi.rr.com> wrote: > > > >> "Virgil" <virgil(a)comcast.net> wrote in message > >> news:virgil-A44A2E.01004230092006(a)comcast.dca.giganews.com... > >> > In article <YDmTg.25600$QT.1073(a)tornado.rdc-kc.rr.com>, > >> > "Poker Joker" <Poker(a)wi.rr.com> wrote: > >> > > >> >> "Virgil" <virgil(a)comcast.net> wrote in message > >> >> news:virgil-9C1609.21071129092006(a)comcast.dca.giganews.com... > >> >> > >> >> > It is in mathematics. Once a proof for any list is established, it > >> >> > covers every list. > >> >> > >> >> This list doesn't contain 4: > >> >> > >> >> 1 > >> >> 2 > >> >> 3 > >> >> > >> >> Proof: > >> >> > >> >> The 1st number isn't 4. > >> >> The 2nd number isn't 4. > >> >> The 3rd number isn't 4. > >> >> That list does't contain 4 > >> >> > >> >> Therefore, Virgil believes that in mathematics, no > >> >> list contains 4. > >> > > >> > As it is PJ's proof, it must be PJ's theorem. > >> > I lay no claim to other's works. > >> > >> How could you? You've never done any work. > > > > Never claimed to have done any. But I can and have appreciated the good > > work of others, which PJ does not. > > Virgil is the jerk who admits to doing nothing That I do not claim to have done anything does not imply that I have not done anything. That PJ does not understand that is a mark of his own arrogance.
From: Virgil on 2 Oct 2006 22:55 In article <rOhUg.28090$8_5.9565(a)tornado.rdc-kc.rr.com>, "Poker Joker" <Poker(a)wi.rr.com> wrote: > "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote in message > news:J6FJxy.FDu(a)cwi.nl... > > In article <YDmTg.25600$QT.1073(a)tornado.rdc-kc.rr.com> "Poker Joker" > > <Poker(a)wi.rr.com> writes: > > > > > > "Virgil" <virgil(a)comcast.net> wrote in message > > > news:virgil-9C1609.21071129092006(a)comcast.dca.giganews.com... > > > > > > > It is in mathematics. Once a proof for any list is established, it > > > > covers every list. > > > > > > This list doesn't contain 4: > > > > > > 1 > > > 2 > > > 3 > > > > > > Proof: > > > > > > The 1st number isn't 4. > > > The 2nd number isn't 4. > > > The 3rd number isn't 4. > > > That list does't contain 4 > > > > > > Therefore, Virgil believes that in mathematics, no > > > list contains 4. > > > > No. You are confusing "some list" with "any list". Understandable if you > > do not know mathematical terminology. > > Obviously you are just learning terminology, so to help you, I'm > providing a definition of "any" for you. The context of "list" in which "any list" occured required such lists to be functions from the naturals to the reals, which The Poker's pseudolists are not. Ergo, the Poker is committing the fallacy of the straw man, which echos the contents of his head quite well.
From: Dik T. Winter on 2 Oct 2006 22:54
In article <1159710187.186119.102420(a)i3g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: And you suggesting to answer in this thread which is just as populated as the other thread... > Dik T. Winter schrieb: > > > In article <1159611066.767146.101490(a)e3g2000cwe.googlegroups.com> muecken= > h(a)rz.fh-augsburg.de writes: .... > > The situation is slightly different. Neither W nor ~W can be proven, at > > least, so mathematicians think. > > Zermelo was not a mathematician? He proved by what today is known as > ZFC: Yup. Assuming the axiom of choice. (Note the 'C' in 'ZFC'.) He did show that W was equivalent to C. Now what? He did *not* show that ZF was equivalent to ZFC. > By forcing it can be proved that, even including AC, the reals cannot > be well ordered. And if one had no proof of that, then experience would > enforce this conclusion. Enlighten me. It can be proven that the axiom of choice is equivalent to the well-ordering theorem (and I think that Zermelo did just that). > > Morover, the negation can be taken in > > two forms leading to elliptical and hyperbolic geometry. Similar with > > ~W. You can take c = aleph_n for any n. > > The situation here is different. The axioms are the same in both cases, > in particular AC is present. It is the same as if in *Euclidean* > geometry the existence of two parallels through one point could not be > be excluded. Eh? W is equivalent to C. So because there is C we have W? Yes. That is trivial. In the same way, the statement "the three angles of a triangle add up to form a straight line" is equivalent to the parallel postulate. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |