Cantor Confusion
in [Math]
|
From: Han de Bruijn on 1 Mar 2007 07:12 Franziska Neugebauer wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > >>On 1 Mrz., 01:22, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: >> >>>In article <1172650983.733643.316...(a)m58g2000cwm.googlegroups.com> >>>mueck...(a)rz.fh-augsburg.de writes: >>> >>>What is misleading about it if every mathmatician calls them >>>"numbers"? >> >>It veils the physical restrictions. > > There is no objective in mathematics to "unvail" physical (material) > restrictions of the world. This makes it possible that common mathematics produces stuff which is properly called "nonsense" in common speech. You have no choice. There is no way around reality. Han de Bruijn
From: Franziska Neugebauer on 1 Mar 2007 07:54 Han de Bruijn wrote: > Franziska Neugebauer wrote: >> mueckenh(a)rz.fh-augsburg.de wrote: >>>On 1 Mrz., 01:22, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: >>>It veils the physical restrictions. >> >> There is no objective in mathematics to "unvail" physical (material) >> restrictions of the world. > > This makes it possible that common mathematics produces stuff which is > properly called "nonsense" in common speech. You have no choice. There > is no way around reality. Mathematicians do not claim to "produce stuff" which is physically (materially) useful in the first place. You are free to call things you do not understand or dislike by whatever name you want. How you call those things is not a mathematical issue and hence off topice in sci.math. F. N. -- xyz
From: Han de Bruijn on 1 Mar 2007 10:37 Franziska Neugebauer wrote: > Han de Bruijn wrote: > >>Franziska Neugebauer wrote: >> >>>mueckenh(a)rz.fh-augsburg.de wrote: >>> >>>>On 1 Mrz., 01:22, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > >>>>It veils the physical restrictions. >>> >>>There is no objective in mathematics to "unvail" physical (material) >>>restrictions of the world. >> >>This makes it possible that common mathematics produces stuff which is >>properly called "nonsense" in common speech. You have no choice. There >>is no way around reality. > > Mathematicians do not claim to "produce stuff" which is physically > (materially) useful in the first place. You are free to call things you > do not understand or dislike by whatever name you want. How you call > those things is not a mathematical issue and hence off topice in > sci.math. Check out 'sci.math' for the keyword "nonsense" and be surprised. Do you deny that Mathematics is supposed to be a no-nonsense discipline? Han de Bruijn
From: Randy Poe on 1 Mar 2007 12:00 On Mar 1, 10:37 am, Han de Bruijn <Han.deBru...(a)DTO.TUDelft.NL> wrote: > Franziska Neugebauer wrote: > > Han de Bruijn wrote: > > >>Franziska Neugebauer wrote: > > >>>mueck...(a)rz.fh-augsburg.de wrote: > > >>>>On 1 Mrz., 01:22, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > >>>>It veils the physical restrictions. > > >>>There is no objective in mathematics to "unvail" physical (material) > >>>restrictions of the world. > > >>This makes it possible that common mathematics produces stuff which is > >>properly called "nonsense" in common speech. You have no choice. There > >>is no way around reality. > > > Mathematicians do not claim to "produce stuff" which is physically > > (materially) useful in the first place. You are free to call things you > > do not understand or dislike by whatever name you want. How you call > > those things is not a mathematical issue and hence off topice in > > sci.math. > > Check out 'sci.math' for the keyword "nonsense" and be surprised. > > Do you deny that Mathematics is supposed to be a no-nonsense discipline? You're playing a semantic game. In math, "nonsense" means that it doesn't make logical sense, that perhaps you've made a claim which does not follow from deduction, or you've made a statement which has no meaning. "Sense" in mathematics doesn't mean "describes a physical object". Lewis Carroll, a mathematician, was fond of playing these semantic games but that doesn't make it mathematics. For example (IIRC): 1. Tyrannical governments are arbitrary. 2. Let A be an any arbitrary government. 3. Then A is tyrannical. That's only a rough approximation. Carroll's "syllogism" was a lot more clever. - Randy
From: mueckenh on 1 Mar 2007 13:41
On 1 Mrz., 01:48, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <1172651370.180739.279...(a)8g2000cwh.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > > On 28 Feb., 02:44, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > In article <1172612805.121491.277...(a)q2g2000cwa.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > > > On 26 Feb., 01:44, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > > > Where? The only reasonably think to do is: > > > > > lim{n -> oo} (2n - |{2,4,5,...,2n}|) = limt{n -> oo} (2n - n) = > > > > > = lim{n -> oo} n. > > > > > So how do you come at the idea that it drops to 0? > > > > > > > > You know that every natural number like 2n is finite and therefore > > > > less than aleph_0 while |{2,4,5,...,2n}| in the limit is aleph_0. > > > > > > And 2n in the limit is aleph_0, when you define such limits. > > > > That is what I say, but according to set theory it is wrong. > > Where is that stated? Set theory does not define limits, so it is not > stated in set theory. Every nondecreasing sequence of real numbers bounded from above has a limit. There is a limit superior and a limit inferior proven by set theory. In order to get to higher ordinals, Hrbacek aqnd Jech say: It is easy to continue the process after this "limit" step is made. > > N is the > > infinite set of all *finite* numbers. Accordinfg to set theory there > > is no infinite number in it although the limit aleph_0 of numbers is > > defined by set theory. > > No, that is *not* defined as "limit". In fact there are limits in set theory. "Es ist sogar erlaubt, sich die neugeschaffene Zahl omega als Grenze zu denken, welcher die Zahlen nu zustreben, wenn darunter nichts anderes verstanden wird, als daß omega die erste ganze Zahl sein soll, welche auf alle Zahlen nu folgt, d. h. größer zu nennen ist als jede der Zahlen nu." > It is stated that from the > axiom of infinity the set N does exist (and does contain finite > elements only, because of the way it is defined). It further > *defines* the cardinality of that set as aleph-0. It does not say > anything about a limit at all. Only because you clever fellows recognized that limits lead to contradictions you abolished any sensible meaning of omega and let it "exist" in the vacuum with no relation to natural numbers. > > > > So in the > > > same way I could state: > > > You know that |{2,4,5,...,2n}| is finite and therefore less than > > > aleph_0, while 2n in the limit is aleph_0. > > > You are actually "proving" here that lim{n -> oo} n/n = 0. Bizarre. > > > > Yes, it is a mess. But it is exactly what everybody asserts when being > > talking about the infinite set of finite numbers. > > You are making a mess of it because of lack of understanding. I am in a good company. But as you even outperform Fraenkel and Levy et al. I cannot cope with you. > Assume > we define limits of sequences of natural numbers such that if the elements > grows without bound the limit is aleph-0. But if we do that, the old > theorem: > lim [ a_n / b_n ] = lim a_n / lim b_n if both limits exist and lim b_n != 0 > no longer holds, which you assume. And that is easy to show, because in > the proof of the theorem use is made of the fact that all limits are > finite (in the context of that theorem a limit exists only if it is finite). > So in the context of the extended definition we have to adjust the condition > on the limit to: > if both limits exist and are finite and lim b_n != 0. > You are making a mess of it because you use a theorem unchanged while a > basic definition has been changed.^ I think Cantor started this game. > > > > And there is the saying (Extensionalitätsaxiom) that a set which is > > > > different from another set must prove this by at least one element. > > > > > > Yes? Where is that in contradiction with what I did state? Give me the > > > set of naturals and any finite set of naturals and I will show you an > > > element where they differ. > > > > Give me Cantors diagonal constructed up to any digit d_nn and I will > > show you the place where d_nn can be fuond in the list. > > > > In case of Cantor's argument you apply the principle: "Compare with > > all numers simultaneously." In case of N differing from its final > > segments you allow only comparison one after the other. That is bad > > logic. > > The comparisons can be done all together, but they will not all yield the > same result, which is not needed, they yield all yield a result, and that > is needed. If the comparison can be done all together, then there must be one result (for linear sets). > > > > > Let S differ by a finite element from all the finite sets A_1, A_2, > > > > A_3, ..., A_n_1. T there is a set A_n, which contains this element, > > > > because every finite element is contained in some finite set A_n. So > > > > there is A_n and infinitely many sets A_n+1, A_n+2, .... which are not > > > > different from S by this element. Further it is clear that a finite > > > > set cannot have an element which is not contained in S. Therefore S > > > > cannot be distinguished by any finite element n from all the finite > > > > sets. > > > > > > Right. > > > > > > > If S exists and is different from any finite set, it must contain > > > > an infinite element w as this is not contained in any finite set. > > > > > > Wrong. Show a proof of this. > > > > You say it above. > > No. I do not say that. I say that S can not be distinguished by any finite > elemen n from all the finite sets. This does inhibit S to exist and being > different from all finite sets, that is why I ask for a proof. It does inhibit S to exist. In fact, S leads to a contradiction if postulated by axiom. > > > > This is you perennial quantifier dislexia. Required: > > > forall A thereis x such that: x notin A and x in S > > > what you state is: > > > thereis x such that forall A: x notin A and x in S > > > the two are *different*. Consider: > > > A1 = {1, 2, 3, 4} > > > A2 = {1, 2, 3, 5} > > > A3 = {1, 2, 4, 5} > > > A4 = {1, 3, 4, 5} > > > A5 = {2, 3, 4, 5} > > > and > > > S = {1, 2, 3, 4, 5} > > > S is different from all of A1 to A5, but there is not a single element > > > where it differs from all of A1 to A5. > > > > Why do you use such ridiculous examples to demonstrate your "non- > > quantifier dyslexia"? > > What is ridiculous about it? These are not linear sets It clearly shows that for a set S to exist > and being different from a collection of other sets it is *not* necessary > that there is a single element where it differs from all those other sets. But this is necessary for linear sets, i.e, such sets which obey A1 c A2 c A3 c ... Regards, WM |