Cantor Confusion
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From: William Hughes on 2 Oct 2006 08:42 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. (i.e. if the second eye is blind you either go with the first eye or decide you can't see anything) - William Hughes
From: Tony Orlow on 2 Oct 2006 09:30 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. > > -William Hughes Oh. I thought it was common sense that the set of even naturals was infinite, so if you remove the even naturals from the vase, it would be empty? What happened to the odd ones? Tony
From: Han de Bruijn on 2 Oct 2006 10:23 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. So far so good for common sense. Mathematics learns us that the limit for time -> noon does not exist. And hence "noon" is a nonsense concept in the context of this problem: the problem becomes ill-posed at noon. Han de Bruijn
From: William Hughes on 2 Oct 2006 10:34 Tony Orlow 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. > > > > -William Hughes > > Oh. I thought it was common sense that the set of even naturals was > infinite, so if you remove the even naturals from the vase, it would be > empty? What happened to the odd ones? Common sense tells you both that the vase is empty and the vase is not empty. There is no "common sense" answer to this problem. - William Hughes P.S. Your question can only be answered from a mathematical perspective. It is true that if you only take even balls from the vase there will be an infinite number of balls left. The answer depends not only on how many balls you take but which balls you take. You agree that at noon there is no balls corresponding to any finite natural number in the vase. Thus, either a ball not corresponding to a finite natural number is put in the vase, or the vase is empty. Now what is the solution to the problem if you specify that only balls corresponding to finite natural numbers are put in the vase?
From: William Hughes on 2 Oct 2006 10:38
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"). "Consistency is all I ask" -Tom Stoppard -William Hughes |