Cantor Confusion
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From: William Hughes on 1 Oct 2006 10:59 mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > Let's refrase it Cantor's way, please: > > (m, m, m, m, m, ...) > > (w, m, m, m, m, ...) > > (w, w, m, m, m, ...) > > (w, w, w, m, m, ...) > > there is no element of the list that contains w's only. But the > > diagonal constructed contains w's only. > > 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. If logic and intuition clash, abandon logic. - William Hughes
From: mueckenh on 1 Oct 2006 11:12 William Hughes schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > Dik T. Winter schrieb: > > > > > > > Let's refrase it Cantor's way, please: > > > (m, m, m, m, m, ...) > > > (w, m, m, m, m, ...) > > > (w, w, m, m, m, ...) > > > (w, w, w, m, m, ...) > > > there is no element of the list that contains w's only. But the > > > diagonal constructed contains w's only. > > > > 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. > > If logic and intuition clash, abandon logic. > The motto of set theorists? Look, instead of the thought experiment constructed by Han and Tony you could also make the following thought experiment: Put 9 balls in the vase and put one ball in the urne. Logic says that the result cannot be different. If logic and set theory clash, abandon set theory. Regards, WM
From: Dave L. Renfro on 1 Oct 2006 11:16 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). Dave L. Renfro
From: William Hughes on 1 Oct 2006 13:08 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > Dik T. Winter schrieb: > > > > > > > > > > Let's refrase it Cantor's way, please: > > > > (m, m, m, m, m, ...) > > > > (w, m, m, m, m, ...) > > > > (w, w, m, m, m, ...) > > > > (w, w, w, m, m, ...) > > > > there is no element of the list that contains w's only. But the > > > > diagonal constructed contains w's only. > > > > > > 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. > > > > If logic and intuition clash, abandon logic. > > > The motto of set theorists? > > Look, instead of the thought experiment constructed by Han and Tony you > could also make the following thought experiment: Put 9 balls in the > vase and put one ball in the urne. Logic says that the result cannot be > different. > No, this is a very different thought experiment. Logic says the result will be different. It matters not only how many balls are added/removed, but also which balls. > 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. -William Hughes
From: Han.deBruijn on 1 Oct 2006 13:42
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. Han de Bruijn |