Cantor Confusion
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From: Han de Bruijn on 2 Oct 2006 10:44 William Hughes wrote: > Han de Bruijn wrote: > >>William Hughes wrote: >> >>>Han de Bruijn wrote: >>> >>>>William Hughes wrote: >>>> >>>>>William Hughes wrote: >>>>> >>>>>>Han.deBruijn(a)DTO.TUDelft.NL wrote: >>>>>> >>>>>>>William Hughes wrote: >>>>>>> >>>>>>>>Han.deBruijn(a)DTO.TUDelft.NL wrote: >>>>>>>> >>>>>>>>>William Hughes schreef: >>>>>>>>> >>>>>>>>>>mueckenh(a)rz.fh-augsburg.de wrote: >>>>>>>>>> >>>>>>>>>>>If logic and set theory clash, abandon set theory. >>>>>>>>>> >>>>>>>>>>Indeed, but logic and set theory do not clash. >>>>>>>>>>Set theory and intuition about infinite sets >>>>>>>>>>clash. >>>>>>>>> >>>>>>>>>Then abandon _both_ (formal / mathematical) logic _and_ set theory. >>>>>>>> >>>>>>>>We are left with intuition. Fine. Oh by the way >>>>>>>>we are going to use my intuition. If you don't like >>>>>>>>it, too bad. Only I can tell what my intuition is. >>>>>>> >>>>>>>Better read better the add-on between parentheses. >>>>>>>We are left with common speech logic and no set theory. >>>>>> >>>>>>O.K. Call it what you want. But if you and I do not >>>>>>agree on what common sense is, I am right. >>>>> >>>>>For example: It's common sense that you can't put more >>>>>than an infinite number of balls in a vase. So if you remove >>>>>an infinite number of balls there are no balls left. >>>> >>>>I don't think there is any common sense about the infinite. >>> >>>[No? Really? "Common sense" is vague feeling based on experience. >>>Why would there be no common sense on something no one >>>has experience with?] >>> >>>O.K. then stop trying to answer the question. If there is no >>>common sense, then there is no common sense answer. >> >>>If both "there are an infinite number of balls in the vase at noon" >>>and "there are no balls in the vase at noon" are equally >>>good (or equally bad) common sense answers, >>>you can't argue that the set theoretic answer is not >>>the correct common sense answer. >> >>Common sense can only answer questions about the finite. It concludes >>that the number of balls becomes larger and larger as we come closer >>to noon. So extrapolating to noon itself can not result in zero balls. >> > Let's see. You start by saying "Common sense can only answer > questions about the finite." You then use common sense to answer > a question about the infinite ("extrapolating to noon"). Yes. That's all we can do with common sense: extrapolating / guessing about the infinite. That's why I added the second part, which has been deleted by you so hastly. Han de Bruijn
From: William Hughes on 2 Oct 2006 10:51 Han de Bruijn wrote: > William Hughes wrote: > > > Han de Bruijn wrote: > > > >>William Hughes wrote: > >> > >>>Han de Bruijn wrote: > >>> > >>>>William Hughes wrote: > >>>> > >>>>>William Hughes wrote: > >>>>> > >>>>>>Han.deBruijn(a)DTO.TUDelft.NL wrote: > >>>>>> > >>>>>>>William Hughes wrote: > >>>>>>> > >>>>>>>>Han.deBruijn(a)DTO.TUDelft.NL wrote: > >>>>>>>> > >>>>>>>>>William Hughes schreef: > >>>>>>>>> > >>>>>>>>>>mueckenh(a)rz.fh-augsburg.de wrote: > >>>>>>>>>> > >>>>>>>>>>>If logic and set theory clash, abandon set theory. > >>>>>>>>>> > >>>>>>>>>>Indeed, but logic and set theory do not clash. > >>>>>>>>>>Set theory and intuition about infinite sets > >>>>>>>>>>clash. > >>>>>>>>> > >>>>>>>>>Then abandon _both_ (formal / mathematical) logic _and_ set theory. > >>>>>>>> > >>>>>>>>We are left with intuition. Fine. Oh by the way > >>>>>>>>we are going to use my intuition. If you don't like > >>>>>>>>it, too bad. Only I can tell what my intuition is. > >>>>>>> > >>>>>>>Better read better the add-on between parentheses. > >>>>>>>We are left with common speech logic and no set theory. > >>>>>> > >>>>>>O.K. Call it what you want. But if you and I do not > >>>>>>agree on what common sense is, I am right. > >>>>> > >>>>>For example: It's common sense that you can't put more > >>>>>than an infinite number of balls in a vase. So if you remove > >>>>>an infinite number of balls there are no balls left. > >>>> > >>>>I don't think there is any common sense about the infinite. > >>> > >>>[No? Really? "Common sense" is vague feeling based on experience. > >>>Why would there be no common sense on something no one > >>>has experience with?] > >>> > >>>O.K. then stop trying to answer the question. If there is no > >>>common sense, then there is no common sense answer. > >> > >>>If both "there are an infinite number of balls in the vase at noon" > >>>and "there are no balls in the vase at noon" are equally > >>>good (or equally bad) common sense answers, > >>>you can't argue that the set theoretic answer is not > >>>the correct common sense answer. > >> > >>Common sense can only answer questions about the finite. It concludes > >>that the number of balls becomes larger and larger as we come closer > >>to noon. So extrapolating to noon itself can not result in zero balls. > >> > > Let's see. You start by saying "Common sense can only answer > > questions about the finite." You then use common sense to answer > > a question about the infinite ("extrapolating to noon"). > > Yes. That's all we can do with common sense: extrapolating / guessing > about the infinite. Nope. We can answer nothing about the infinite using common sense. Extrapolation to the infinite is not a common sense operation. (Clearly it does not make sense to do anything an infinite number of times, and it makes even less sense to talk about what happens after we do something an infinite number of times.) -William Hughes
From: Randy Poe on 2 Oct 2006 11:02 Poker Joker wrote: > "Randy Poe" <poespam-trap(a)yahoo.com> wrote in message > news:1159578269.577169.76000(a)m73g2000cwd.googlegroups.com... > > >> 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. > > > > That doesn't make sense. You are saying that every digit of r > > both is equal to 4 and is equal to 5. > > So when it's put in extremely simple terms, then you understand > that the process doesn't always make sense. No, I understand that you wrote something that doesn't make sense, but that also bears no resemblance to the proof you're confused about. Nowhere does anybody sane say that there is a digit which is simultaneously required to have two different values, in any variant of the Cantor proof (except yours). - Randy
From: mueckenh on 2 Oct 2006 13:58 Dave L. Renfro schrieb: > Peter Webb wrote (in part): > > >> This is a complete red herring. There is no question that > >> the Real generated by Cantor's proof is computable (r. e,) > >> if the original list is, [...] > > mueckenh(a)rz.fh-augsburg.de wrote (in part): > > > Of course. That's why the diagonal proof only proves the > > existence of numbers which belong to a countable set i.e. the > > set of constructible reals. This proof proves in essence that > > the countable set of constructible real numbers is uncountable. > > A fine result of set theory. > > You're overlooking Peter Webb's hypothesis "if the original > list is". You need to have a list (x_1, x_2, x_3, ...) such > that the function given by n --> x_n is computable. Thus, > before you can conclude what you're saying (which sounds like > a metalogic "proof by contradiction" to me, but no matter), > you need to come up with a computable listing of the computable > numbers (or at least, show that such a listing exists). One cannot compute a list of all computable numbers. By this definition, (1) the computable numbers are uncountable. (2) There is no question, that the computable numbers form a countable set. This is a contradiction. It is not necessary to come up with a list of all computable numbers. Regards, WM
From: Virgil on 2 Oct 2006 14:04
mueckenh(a)rz.fh-augsburg.de wrote: > This is the typical one-eyed view of a set theorist. The same we have > with Han's vase: Of course there is no ball which has not jumped out at > noon. We cannot name any such number. But the other eye should see that > there are more balls in than out at any time, including noon. "Mueckenh" again shows himself to be cross-eyed. |