Cantor Confusion
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From: mueckenh on 11 Oct 2006 05:59 Virgil schrieb: > In article <b7f51$452a1029$82a1e228$25909(a)news2.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > Mathematicians have found another name for scientitic facts. They call > > them "just an opinion". > > Mathematicians do not contest the alleged factualness of scientific > "facts", but do contest their relevance in determining what > mathematicians are to be allowed to think. > Who defined what they are allowed to think? How came logic to makind? By observing what happens in our physical reality. If you dismiss it, why don't you use some other kind of logic? There are rules obtained from reality. If you want to use them then you should also use axioms of mathematics obtained from reality. Regards, WM
From: mueckenh on 11 Oct 2006 06:10 William Hughes schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > > > > > In my view we have not gotten very far. We still have > > > > > the result that there is no list of all real numbers > > > > > > > > That is not astonishing, because there are only those few real numbers > > > > which can be constructed. > > > > > > Few? Few compared to what. > > > > Compared to the assumed set of uncountably many. > > Funny, you claim that the term "uncountably many" has no > meaning, but you use it. You believe in its meaning and in the great set R. > > > > > > The real numbers that cannot > > > be constructed? According to you they don't exist. But even > > > these "few" real numbers cannot be listed! > > > > Nevertheless the diagonal proof shows only that there are elements of a > > countable set which have not yet been constructed.\ > > No, it is much stronger. It shows that any list of constructable > numbers > is not complete. Because it had not been constructed. Nevertheless it shows that the constructed number belongs to a countable set. Therefore all can be put in bijection with N --- after the conxtruction is complete. > > > > > > > > > > > > > > (we need to reinterpret our terms, real numbers are > > > > > computable real numbers, and a list is a computable > > > > > function from the natural numbers to the (computable) real > > > > > numbers). > > > > > > > > > > If it gives you a warm fuzzy to say that > > > > > "Every ball will be removed at some time before noon", > > > > > > > > No. To say that every ball will be removed, is wrong, because there is > > > > not every ball. > > > > > > > > > > If it gives you a warm fuzzy to say > > > > > > "For any natural N, the ball numbered N will be removed from > > > the vase before noon" > > > > There is not "any natural" but only those which we can define. > > O, so there are now not only now but always > only a finite number of naturals, not even > an arbitrarially large number. But you continue to > prattle on about limits. > > > There is > > a largest natural which ever will be defined. Hence mathematics in the > > universe and in eternity has to do with only a very small sequence of > > naturals. > > > > Writing 1,2,3,... is but cheating > > > > If you want to deal with a system in which there is an unknown > but largest natural, knock yourself out. That is nonsense. There is no largest natural! There is a finite set of arbitrarily large naturals. The size of the numbers is unbounded. > But you have a long > way to go before you are even close to being consistent. It is just the reality. It is impossible to have more than 10^100 numbers represented by all the bits of the universe. > And don't attempt to use results from this system to say that > results from another system are wrong. > > Note that according to you the ball in vase problem > is trivial. At some time, strictly before noon we will > reach the largest natural. After this, nothing happens. There is no largest natural, but certainly the vase problem is trivial, because there are less than 10^100 balls. This discssion is only interesting in order to find internal contradictions of set theory. That set theory is completely incapable to describe anything correctly with regards to reality, that is obvious. Regards, WM
From: mueckenh on 11 Oct 2006 06:15 Virgil schrieb: > In article <1160397914.738238.238220(a)m7g2000cwm.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > > Note, the question originally asked was very careful to > > > distinguish between the questions " Will the whole autobiography > > > be written?", and "Will certain pages of the autobiography > > > be written?, so my repharasing is accurate. > > > > Yes, but the assertion of Fraenkel and Levy was: "but if he lived > > forever then no part of his biography would remain unwritten". That is > > wrong, because the major part remains unwritten. > > What part? That part accumulated to year t, i.e., 364*t. If you think Lim {t-->oo} 364*t = 0, we need not continue to discuss. Regards, WM
From: mueckenh on 11 Oct 2006 06:33 Dik T. Winter schrieb: > In article <1160407188.555466.59730(a)i3g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > ... > > > > The limit {1,...,n} for n-->oo is N, if N does exist. > > > > > > By what definitions? > > > > By the definition of the limit ordinal omega (= N). > > Pray, explain. The ordinal omega is the smallest ordinal larger than each > finite ordinal. That is the definition. How do you come at the above > limit from that? > By Cantor's definition. Cantor introduced omega as a limit aordinal number. p. 325: "Die zweite Zahlenklasse hat eine kleinste Zahl omega = Lim n." Cp. p. 328 and many others of his collected works. "Lim[n] n = omega." What else should it be? > What *might* be a sensible definition of a limit for a sequence of sets of > naturals is, that (given each A_n is a set of naturals), the limit > lim{n = 1 ... oo} A_n = A Yes, in that manner the definition runs. Cantor does not write n = 1...oo but puts only the n (he uses nue) under the limit. But the meaning is clearly this one. > exists if and only if for every p in n, there is an n0, such that either > (1) p in A_n for n > n0 > or > (2) p !in A_n for n > n0. > In the first case p is in A, in the second case p !in A. > > With that definition, indeed, > lim{n = 1 ... oo} A_n = N, > but also > lim{n = 1 ... oo} {n + 1, ..., 10n} = 0. > > I do not think you are meaning that definition. So what *is* your > definition? I do *not* believe that omega exists or is a useful notion. Therefore I do not give a definition, but, if necessary during the discussion, I use the only posdsible one as given by Cantor (see above). Regards, WM
From: mueckenh on 11 Oct 2006 06:37
Dik T. Winter schrieb: > In article <1160405521.639470.295030(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > > > In article <virgil-F7008E.15353808102006(a)comcast.dca.giganews.com> Virgil <virgil(a)comcast.net> writes: > > > ... > > > > If one chooses to work within ZFC or NBG, the vase is empty at noon. > > > > > > I doubt this. The problem is not defined with enough precision to state > > > that. It has not been defined by most what is meant with "the number of > > > balls in the vase at noon". Of course, you can use that the infinite > > > intersection of sets does exist (and that is what you are using), and > > > so get at the result. > > > > Then the infinite intersection of the cardinal numbers A(t) with t = 1, > > 2, 3, ... of the set in the vase after completing action t does also > > exist. It is 9. > > Impossible. The infinite intersection is a set. Every cardinal is a set. so 9 is a set. If you don't like this, then apply: The intersection of the cardinal numbers has the cardinal number 9. Regards, WM |