Another AC anomaly?
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From: Virgil on 11 Dec 2009 15:24 In article <03e1afc6-ec37-4212-b958-063a237d2bb4(a)f16g2000yqm.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 11 Dez., 03:28, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article > > <89fb6e91-b6b1-4926-afca-820492e3c...(a)r24g2000yqd.googlegroups.com> WM > > <mueck...(a)rz.fh-augsburg.de> writes: > > �> On 10 Dez., 15:40, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > �> > > �> > Have a look at <http://en.wikipedia.org/wiki/Lim_inf> in the section > > �> > titled "Special case: dicrete metric". =A0An example is given with the > > �> > sequence {0}, {1}, {0}, {1}, ... > > �> > where lim sup is {0, 1} and lim inf is {}. > > �> > > > �> > Moreover, in what way can a definition be invalid? > > �> > > �> It can be nonsense like the definition: Let N be the set of all > > �> natural numbers. > > > > In what way is it nonsense? �Either that set does exist or it does not > > exist. > > If it does exist there is indeed such a set, if it does not exist there is > > no set satisfying the definition. �In both cases the definition is not > > nonsense in itself. > > It is nonsense to define a pink unicorn. It is not nonsense to define a thing, though it may be nonsense to insist that such a definition is instanciated. > The set N does not exist as > the union of its finite initial segments. It does in ZF and in other sensible places outside of Wolkenmuekenheim. This is shown by the (not > existing) path 0.000... in the binary tree. Again, in Wolkenmuekenheim there does not exist an complete infinite binary tree, though it does exist in ZF and other set theories that one can find outside of Wolkenmuekenheim. > > Let {1} U {1, 2} U {1, 2, 3} U ... = {1, 2, 3, ...}. > What then is > {1} U {1, 2} U {1, 2, 3} U ... U {1, 2, 3, ...} ? > If it is the same, then wie have a stop in transfinite counting. {1} U {1, 2} U {1, 2, 3} U ... U {1, 2, 3, ...} is ambiguous, so is meaningless. > If it is not the same, what is it? The expression, being ambiguous, has no meaning. On the other hand, ({1} U {1, 2} U {1, 2, 3} U ...) U {1, 2, 3, ...} does have meaning and equals {1,2,3,...} > > > > But apparently you are of the opinion that you are only allowed to define > > things that do exist. > > Most essential things in mathematics exist without definitions and, > above all, without axioms. Nothing in mathematics can exist without some a priori assumptions about its properties, which constitute the axioms WM would dispense with. So that in WM's world, there are all these things floating around with no known or knowable properties. At least in Wolkenmuekenheim
From: Virgil on 11 Dec 2009 15:32 In article <3be6133f-3000-429e-a8be-b51d5f575fd9(a)m16g2000yqc.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 11 Dez., 08:03, "K_h" <KHol...(a)SX729.com> wrote: > > > It depends on the context. �When it comes to supertasks, > > limsup={0,1} is basically useless. �That is why those > > definitions are not good, and invalid, for evaluating > > supertasks -- in response to WM's supertask issues. > > Why do you think that the supertask of uniting all natural numbers is > not invalid? Does WM find it impossible to distinguish between natural numbers and objects which are not natural numbers? That impossibility is a natural and inevitable consequence of not having a set of natural numbers. Being a set, in general, merely establishes a boundary between what is to be in the set and what is to be outside of it. If nothing is indeterminant with respect to being inside or outside, then the boundary defines a set.
From: Virgil on 11 Dec 2009 15:52 In article <364fe723-5ae4-4ed5-9219-c6b3892b3df2(a)d21g2000yqn.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > I did not say that there is a final step. I say that there is no > chance for a difference of lim card(S_n) and card(lim(S_n)) where lim > means n --> oo. But "lim" requires a deal more than merely "n --> oo". In each case, "lim_[n --> oo] card(S_n)" and "card( lim_[n --> oo](S_n))", one has to say under what conditions the alleged limit will actually exist and how its value is determined when it does exist. And with Dik's definitions, the two limits need not be equal.
From: Virgil on 11 Dec 2009 15:53 In article <364fe723-5ae4-4ed5-9219-c6b3892b3df2(a)d21g2000yqn.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > If, in your example you would claim that lim(1/n) = 0 and and 1/omega > = 10 That seems to be much more like something that WM would claim than something Dik would claim.
From: Virgil on 11 Dec 2009 16:22
In article <d5a795ed-90bd-43bd-af2f-dd4ab437aca8(a)d21g2000yqn.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 11 Dez., 03:50, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article > > <fec95b83-39c5-4537-8cf7-b426b1779...(a)k17g2000yqh.googlegroups.com> WM > > <mueck...(a)rz.fh-augsburg.de> writes: > > �> On 10 Dez., 16:35, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > �> > �> Before 1908 there was quite a lot of mathematics possible. > > �> > > > �> > Yes, and since than quite a lot of newer mathematics has been made > > �> > available. > > �> > > �> Most of it being rubbish. > > > > Nothing more than opinion while you have no idea what has been done in > > mathematics since 1908. �Algebraic number theory is rubbish? > > The answer is an explicit no. "Most" here concerns the magnitude of > numbers involved. There was much ado about inaccessible cardinals. There has been much ado about all sorts of other things since 1908, enough so that what has been said about cardinals is far from being "most" of it. > > > > �> > Moreover, before 1908 mathematicians did use concepts without actually > > �> > defining them, which is not so very good in my opinion. > > �> > > �> Cantor gave a definition of set. What is the present definition? > > > > Something that satisfies the axioms of ZF (when you are working within ZF). > > It is similar to the concepts of group, ring and field. �Something that > > satisfies those axioms is such a thing. �But I think you find all those > > things rubbish. > > Why that? Group, ring and field are treated in my lessons. Are they treated as badly as sets? > > > > > �> > �> There is not even one single infinite path! > > �> > > > �> > Eh? �So there are no infinite paths in that tree? > > �> > > �> In fact no, but every path that you believe in is also in the tree, > > �> i.e., you will not be able to miss a path in the tree. > > > > I believe in infinite paths, you state they are not in the tree. �So we > > have a direct contradiction to your assertion. > > You believe in infinite paths. But you cannot name any digit that > underpins your belief. Every digit that you name belongs to a finite > path. Every digit belongs to infinitely many finite path-segments, so why not to any infinite path? To get an infinite tree one must have something being infinite, so why does WM claim that there can there be infinitely many paths in the tee but not infinitely many nodes in any path? How is one such actual infiniteness any more actual than any other? > Every digit that is on the diagonal of Canbtor's list is a > member of a finite initial segment of a real number. Actually, any such list is not Cantor's but is presented to him by a challenger, and he is challenged to find an unlisted sequence, which he can always do, by applying a simple rule of construction. > > You can only argue about such digits. On the contrary, we can argue about a good deal more, e.g., sequences of digits. > And all of them (in form of bits) are present in my binary tree. But your tree has no paths (MAXIMAL sequences of bits with each bit having exactly one child bit in the same path). > > > > �> > �> � � � � � � � � � � � � � � � � � But there is every path > > �> > �> which you believe to be an infinite path!! Which one is missing in > > �> > �> your opinion? Do you see that 1/3 is there? > > �> > > > �> > If there are no infinite paths in that tree, 1/3 is not in that tree. > > �> > > �> 1/3 does not exist as a path. But everything you can ask for will be > > �> found in the tree. > > �> Everything of that kind is in the tree. > > > > This makes no sense. �Every path in the tree (if all paths are finite) is > > a rational with a power of 2 as the denominator. �So 1/3 does not exist > > as a path. �In what way does it exist in the tree? > > It exists in that fundamentally arithmetical way: You can find every > bit of it in my binary tree constructed from finite paths only. I can find every letter in any book in an alphabet, but that does not make any alphabet contain the book. > You > will fail to point to a digit of 1/3 that is missing in my tree. You will fail to point to any letter of 'Hamlet' that is missing in my alphabet. > Therefore I claim that every number that exists is in the tree. Therefore I claim that everything in 'Hamlet' is in my alphabet. > > Isn't a path a sequence of nodes, is it? In any tree, if a path has a last node, then that node must be a leaf node having no child nodes. > Everey node of 1/3 (that you > can prove to belong to 1/3) is in the tree. > > > > �Similar for 'pi' and 'e'. > > Yes. Every digit is available on request. > > >�So when you state that > > the number of paths is countable that does not mean that the number of real > > numbers is countable because there are apparently real numbers in your tree > > without being a path. > > Wrong. Not only "apparantly" but provably (on request): Every digit of > every real number that can be shown to exist exists in the tree. Every letter in 'Hamlet' is in my alphabet. > > Or would you say that a number, every existing digit of which can be > shown to exist in the tree too, is not in the tree as a path? 'Hamlet' is not "in" any alphabet, even though every letter in it is in the alphabet. It is not the individual letters so much as the sequence of them that creates 'Hamlet'. Unless every sequence of digits in real number is in a path, that path does not represent that number, and possibly not even when they are. |