Cantor Confusion
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From: mueckenh on 2 Oct 2006 14:24 Virgil schrieb: > 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, daß jede Menge wohlgeordnet werden kann", Math. > > Ann. 59 (1904) 514 - 516 > > Zermelo, E., "Neuer Beweis für die Möglichkeit 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. The clue is: There is a well-order (proven by Zermelo) but we can never know how it looks like (proven by forcing). What is a thing worth, which cannot exist other than in our mind but which provably does not exist in our mind? Regards, WM
From: mueckenh on 2 Oct 2006 14:26 Virgil schrieb: > 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". It is enough to distinguish finite numbers, but not to distinguish infinite numbers. 1 + n =|= n but 1 + omega = omega. Regards, WM
From: mueckenh on 2 Oct 2006 14:28 Virgil schrieb: > > 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, Then obviously it is not logic, what you ever have seen. > > > > If logic and set theory clash, abandon set theory. > > My logic and my set theory get along famously. With that what you call logic, the names of the balls are important. But by bijection, we can exchange 11 and 2 or 21 and 3. So applying the same logic of yours, we see that the names are unimportant. Regards, WM
From: MoeBlee on 2 Oct 2006 14:37 Poker Joker wrote: > I never tried to refute the uncountability of the reals. Good. And you've not correctly refuted diagonal arguments that prove the uncountability of the reals. > I don't need to respond to you about Arturo. Okay, you don't have such a need. Meanwhile, my point stands that your original remarks about Arturo's post are incorrect. MoeBlee
From: Virgil on 2 Oct 2006 15:02
In article <1159813276.334810.303640(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > 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. > > It is one of the worst reproaches at all if one can accuse a theory > that its results depend on the symbols chosen to denote its elements. Then mathematics is immune to that reproach, as all its symbols are completely arbitrary. |