Cantor Confusion
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From: Virgil on 14 Oct 2006 16:58 In article <1160815058.133714.255490(a)i3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <990aa$452e542e$82a1e228$16180(a)news1.tudelft.nl>, > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > > > Completed infinity > > > does not exist. > > > > There are more things in heaven and earth, Horatio, > > Than are dreamt of in your philosophy. > > And the set of those things contains also some impossibilities which > you are not aware of. > > Regards, WM Some of those "impossibilities" are only impossible in "Mueckenh"'s philosophy, and need not be in other philosophies.
From: Virgil on 14 Oct 2006 17:02 In article <1160815134.774717.182680(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1160675140.906009.253460(a)i42g2000cwa.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > > this does not imply > > > > > > > > there exists a single unary number M such that for every digit > > > > position N, M covers 0.111... to position N > > > > > > Why shouldn't it? > > > > In general > > "for all x there is a y such that f(x,y)" > > does not imply > > "there is a y such that for all x f(x,y)". > > > > To establish the latter requires proof over and above the former. > > I did not state that this be true in general, but it is true in a > special case, namely for the covering of linear sets of finite > elements. What does "covering of linear sets of finite elements" mean? The infinite set of naturals, as it exists in ZF and NBG, is a linear set of finite elements according to the usual meanings of "linear order" and "finite elements".
From: Virgil on 14 Oct 2006 17:04 In article <1160815255.198845.303670(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > Han de Bruijn wrote: > > > Dik T. Winter wrote: > > > > In article <1160647755.398538.36170(a)b28g2000cwb.googlegroups.com> > > > > mueckenh(a)rz.fh-augsburg.de writes: > > > > ... > > > > > It is not > > > > > contradictory to say that in a finite set of numbers there need not > > > > > be > > > > > a largest. > > > > > > > > It contradicts the definition of "finite set". But I know that you are > > > > not interested in definitions. > > > > > > Set Theory is simply not very useful. The main problem being that finite > > > sets in your axiom system are STATIC. They can not grow. Which is quite > > > contrary to common sense. (I wouldn't imagine the situation that a table > > > in a database would have to be redefined, every time when a new row has > > > to be inserted, updated or deleted ...) > > > > Is your claim only that set theory is not useful or is contrary to > > common sense? Or, are you claiming something more, e.g., that set theory > > is mathematically inconsistent? > > It is not useful and contrary to common sense but above all it is > mathematically inconsistent. Such outrageous claims require proofs which "Mueckenh" has not been able to provide. That standard set theories may violate "Mueckenh"'s amour propre does not constitute proof. If I remember correctly, you offered to > formulate the vase problem in your language. Perhaps you can see the > contradiction there. > > Regards, WM
From: Virgil on 14 Oct 2006 17:28 In article <1160834055.223900.182920(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1160676113.344404.246370(a)h48g2000cwc.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Dik T. Winter schrieb: > > > > > > > > > > Cantor uses Lim{n} n = omega witout much ado. > > > > > > > > That is not a limit of "sets". > > > > > > It is the limit of the natural numbers. The limit is omega, an ordinal > > > number. Meanwhile we know that every number is a set. Hence, it is a > > > limit of sets. > > > > N ( or omega) is only a limit in the sense of being a union of its > > members, and is the first non-empty ordinal to be equal to the union of > > its members. No other form of 'limit of a sequence' of sets is defined > > in ZF. > > This union just gives Lim {n-->oo} {1,2,3,..,n} = N. If "Lim {n-->oo} {1,2,3,..,n}" is to be given a definition at all, then N would be a useful one, but that is not how N is "generated", at least in ZF or NBG. > > > > > > > > Yes, and that fits my definition. On the other hand, how would you > > > > define lim{n --> oo} {n, n+1, ...}? > > > > > > I would not attempt to define that. > > > > It is obviously the empty set, i.e., the intersection of all the sets of > > form {n, n+1, ...}. > > Above you wrote that only the *union* and "no other form of 'limit of > a sequence' of sets is defined in ZF". Now you accept the > intersection. Given a family of sets F and their union G, for each S in F, one has G\S = {x in G: not x in S}, by the axiom of separation. Let H be the union of all the G\S, then G\H is the intersection of all sets in F. So one does not need an additional definition to get arbitrary intersections, it follows from the definition of union and the axiom scheme of separation. > Regards, WM
From: William Hughes on 14 Oct 2006 17:29
mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > Tony Orlow schrieb: > > > > > > > > > > > > > Why shouldn't it? If every digit position of 0.111... is a finite > > > > > position then exactly this is implied. Your reluctance to accept it > > > > > shows only that you do not understand how an infinite set can consist > > > > > of finite numbers. In fact, nobody can understand it, because it is > > > > > impossible. > > > > > > > > > > Regards, WM > > > > > > > > > > > > > But Wolfgang, surely that consideration does not impact, say, the set of > > > > reals in (0,1], which are all finite, yet whose number is infinite. It > > > > is not a requirement that a set of all finite values be finite. That > > > > conclusion follows from the combination of that fact with the fact there > > > > is a constant positive unit difference between consecutive elements. > > > > > > Of course, Tony, you are right! > > > > But only a finite number of real numbers will every be described in > > the lifetime of the universe. Surely by your reasoning there must be > > a finite number of real numbers? > > Yes, but the reason you mentioned, correctly, is another one than Tony > had in mind, correctly. > TO says that the number of reals in [0,1] is infinite. You say that the number of reals is finite. You say there is no contradiction. - William Hughes > Regards, WM |