Cantor Confusion
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From: Virgil on 1 Oct 2006 14:56 In article <oaPTg.5137$3E2.3874(a)tornado.rdc-kc.rr.com>, "Poker Joker" <Poker(a)wi.rr.com> 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. Joker's processes do not make sense to anyone but Joker. Which is possibly why he calls himself "Joker".
From: William Hughes on 1 Oct 2006 15:03 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. -William Hughes
From: Virgil on 1 Oct 2006 15:12 In article <1159710187.186119.102420(a)i3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1159611066.767146.101490(a)e3g2000cwe.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > cbrown(a)cbrownsystems.com schrieb: > > ... > > > > Therefore, the assertion "M is a complete list of reals" is only true > > > > if the assertion "M is complete, and M is not complete" is true. > > > > > > > > (A and ~A) = false. > > > > > > A system has the property W, if it can be proved that the reals can be > > > well-ordered. A system has the property ~W if it can be proved that the > > > reals cannot be well-ordered. A system is self-contradictive, if W and > > > ~W can be proved. Therefore the system does not exist. > > > > 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: > > Zermelo, E., "Beweis, da? jede Menge wohlgeordnet werden kann", Math. > Ann. 59 (1904) 514 - 516 > Zermelo, E., "Neuer Beweis f?r die M?glichkeit einer Wohlordnung", > Math. Ann. 65 (1908) 107 - 128 > > > So either W or ~W can be taken as a new > > axiom, leading to different branches of set theory. The case is similar > > to the parallel postulate which can not be proven from the other > > postulates, > > so either that postulate or its negation can be taken as an axiom, leading > > to different branches of geometry. > > By forcing it can be proved that, even including AC, the reals cannot > be well ordered. That is not in accord with the following: http://en.wikipedia.org/wiki/ZFC#The_axioms Axiom of choice: For any set X there is a binary relation R which well-orders X. This means that R is a linear order on X and every nonempty subset of X has an element which is minimal under R.
From: Virgil on 1 Oct 2006 15:21 In article <1159710911.446611.96530(a)e3g2000cwe.googlegroups.com>, 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. The > refore set theory is useless. One cannot calculate meaningfully with > infinites! One can do some calculations with "infinites" if one is sufficiently careful. > > To come back to your argument: The diagonal differs from all the list > numbers at most in one w. That is enough to distinguish between any two successive list members, so is enough to distinguish the "diagonal". > If there was not a list number begining with > w for every w (except the first) of the diagonal, then the diagonal > could not exist. On the contrary, whatever pattern of m's and w's occurs in the listed strings, there is a string, misleadingly called the diagonal, which differs from the nth listed string in the nth place, and is, therefore, not in the list. So the diagonal differs in at most one w. That does > not matter, however, because a last w can be shown to be not existing. > > Regards, WM
From: Virgil on 1 Oct 2006 15:25
In article <1159715553.956057.69470(a)i42g2000cwa.googlegroups.com>, 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? On the contrary, the motto of their critics. > > 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. That does not follow from any form of logic I have ever seen, and I have seen a fair amount, > > If logic and set theory clash, abandon set theory. My logic and my set theory get along famously. If "Mueckenh"'s logic and his set theory clash. perhaps he should give up mathematics entirely. |