Cantor Confusion
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From: Virgil on 2 Oct 2006 15:05 In article <1159813454.884946.261560(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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, 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. > > The clue is: There is a well-order (proven by Zermelo) but we can never > know how it looks like (proven by forcing). The axiom of choice requires that it exist in the same sense that any mathematical object exists, > What is a thing worth, which cannot exist other than in our mind but > which provably does not exist in our mind? How much it is worth is quite a different question from whether it exists. > > Regards, WM
From: Virgil on 2 Oct 2006 15:09 In article <1159813617.658456.185930(a)c28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. It is enough to distinguish one function from another: Given f:A ->B and g:A->B ands a in A such that f(a) != g(a) then f != g. In the above, A = N and B = {m,w}, so that differing at one place is enough to make them different functions.
From: Virgil on 2 Oct 2006 15:13 In article <1159813725.103198.240230(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. If you argue that the names are unimportant in the way you say, then you are also arguing that 21 is prime and 11 is even.
From: Tony Orlow on 2 Oct 2006 15:13 mueckenh(a)rz.fh-augsburg.de wrote: > 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, 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. > > 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 > Hi Wolfgang. I don't know if you saw it, but I had quite a long tangle about this in the thread Well-Ordering The Reals", where I introduced the H-riffic numbers. I have to re-publish (like here) that paper and others. It's impossible to explicitly well order the reals, as far as I can tell. I did my best. Tony
From: Tony Orlow on 2 Oct 2006 15:14
mueckenh(a)rz.fh-augsburg.de wrote: > 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 > IFR distinguishes between a set and itself plus one. |