Another AC anomaly?
in [Logic]
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From: Marshall on 11 Dec 2009 13:53 On Dec 11, 9:13 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > On 11 Dez., 03:50, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > > > > > > 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. You > will fail to point to a digit of 1/3 that is missing in my tree. > Therefore I claim that every number that exists is in the tree. This argument can be inverted to "prove" the existence of a natural number whose decimal expansion is an infinite string of 3s. The infinite-3 number exists in a fundamentally arithmetical way: you can find every digit of it in a preceding natural number constructed from finite string of 3s only. You will fail to point to a digit of ...333 that is missing in the natural numbers. Therefore you claim the naturals and the reals are just the same, but in the reverse direction, just as AP says they are. Marshall
From: WM on 11 Dec 2009 14:33 On 11 Dez., 19:53, Marshall <marshall.spi...(a)gmail.com> wrote: > On Dec 11, 9:13 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > > > On 11 Dez., 03:50, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > > > > > > > > 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. You > > will fail to point to a digit of 1/3 that is missing in my tree. > > Therefore I claim that every number that exists is in the tree. > > This argument can be inverted to "prove" the existence of > a natural number whose decimal expansion is an infinite > string of 3s. No. Why should it? Of course the magnitude of natural numbers is not limited. For every number with n digits 333...333 there is another number with n^n digits. Nevertheless each one is finite. > The infinite-3 number exists in a fundamentally > arithmetical way: you can find every digit of it in a preceding > natural number constructed from finite string of 3s only. > You will fail to point to a digit of ...333 that is missing in > the natural numbers. Of course. That is because there is no digit missing in the natural numbers. > Therefore you claim the naturals and > the reals are just the same, but in the reverse direction, > just as AP says they are. They *are* just the same, because your argument that above procedure would prove an infinite string of 3's is wrong. There is neither a natural nor a rational with an infinite string of digits. To be able to determine every digit of a number you like does not imply that there is a number with a never ending sequence of digits. Why should it??? Because Cantor believed that God knows infinite strings? (He read it in civitate dei of St. Augustinus.) Or because Zermelo misunderstood Bolzana-Dedekind's definition of infinity? Are you really thinking that infinity comes into being because a Dr. Zermelo of Germany said so? Regards, WM
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. |