Another AC anomaly?
in [Logic]
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From: WM on 10 Dec 2009 12:18 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. > 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? > > > N need not exist as a set. If n is a natural number, then n + 1 is a > > natural numbers too. Why should sets be needed? > > Ok, so N is not a set. What is it? N is a sequence of natural numbers. > > 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. > > > 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. > Otherwise 1/3 would be a rational with a denominator that is a power of > 2 (each finite path defines such a number). > > > What node of pi is missing in the tree constructed by a countable > > number of finite paths (not even as a limit but by the axiom of > > infinity)? > > By the axiom of infinity there *are* infinite paths in that tree. So your > statement that there are none is a direct contradiction of the axiom of > infinity. Try to find something that exists in your opinion but that does not exist in the tree that I constructed. Regards, WM
From: Virgil on 10 Dec 2009 15:19 In article <6d236085-34a7-4c43-bbe6-341bf8bb6faf(a)l13g2000yqb.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 10 Dez., 08:19, Virgil <Vir...(a)home.esc> wrote: > > In article > > <b963b9aa-c345-43cf-bf78-e9e27401f...(a)c34g2000yqn.googlegroups.com>, > > > > > > > > > > > > �WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > On 10 Dez., 01:58, "K_h" <KHol...(a)SX729.com> wrote: > > > > > > > When using an intermediate reservoir, as shown in my > > > > > lesson > > > > >http://www.hs-augsburg.de/~mueckenh/GU/GU12.PPT#394,22,Folie > > > > > 22 > > > > > it becomes clear that N cannot be generated by adding > > > > > number after > > > > > number. > > > > > > Why not? �Say we have an infinitely large sheet of paper and > > > > we print each natural number, n, on the paper at time > > > > t=1-1/(n+1). �Certainly at time t=1 we have all the naturals > > > > printed on the page. > > > > > It seems so. But it is wrong. You see it if you consider the > > > alternative process using an intermediate reservoir as "realized" in > > > my lesson above. > > > > > naturals - reservoir - paper > > > N � � � � � - �{ } � � � � � �- { } > > > N/{1} � � �- �{1} � � � � �- { } > > > N/{1,2} � �- {2} � � � � � - {1} > > > N/{1,2,3} - {3} � � � � �- {1,2} > > > ... > > > �N/{1,2,3, ...,n} - {n} - {1,2,3, ...,n-1} > > > ... > > > > > The set in the middle contains a number at every time after t = 0. > > > Hence this number cannot yet have been printed on the paper (because > > > it will be printed only after its follower will have entered the > > > reservoir). > > > > And when all the numbers have passed through your "reservoir", both into > > and out ofd it, as will have happened by t = 1, which numbers does WM > > claim will still be unprinted. > > Then a last one would have had to leave the intermediate reservoir. The whole point of infinite series and sequences is that there is no such thing as a "last one". Though when has gone through them all there is an end of the process of going through them. > > > > When going through the terms of an infinite sequence, as in the above, > > EITHER the process hangs up on a particular term of that sequence OR it > > goes through every term of the sequence, TERTIUM NON DATUR.- > > Or it is proved that the assumption of finished infinity is nonsense. But that is exactly what, among many other things, WM has not proved. > Secundum non datur. Then rimum non datur as well, at least in Wolkenmuekenheim.
From: Virgil on 10 Dec 2009 15:23 In article <89fb6e91-b6b1-4926-afca-820492e3cc6d(a)r24g2000yqd.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > 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". �An 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. If there cannot be a set of them, which only requires a way to distinguish them from other things, then there can be no proofs about properties of all of them. E.g., such properties as m + n = n + m for ALL naturals cannot be proved. And thus cannot be proved for all rationals or all reals either.
From: Virgil on 10 Dec 2009 15:33 In article <5333fb9a-1670-4fcc-85d3-25e75fb5bd1d(a)f16g2000yqm.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 10 Dez., 16:29, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > Without the axiom of infinity omega would not be immediately existing. > > So apparently there is a definition of omega without the axiom of infinity. > > Can you state that definition? > > Look into Cantor's papers. Look into my book. Cantors papers might be worth it, but no serous mathematician need bother with any book written by WM. Wm has sufficiently often proved his mathematical incompetence here to obviate any need to delve further into it. > > > > There are no concepts of mathematics without definitions. > > So? What is a set? Does the absence of a definition of "set" imply the absence of all definitions from mathematics? If not then WM's question is, as usual, totally irrelevant. > > > > > �> An infinite union *is* not at all. But if it were, it was a limit. > > > > It *is* according to one of the axioms of ZF, and as such it is not a limit. > > It *was* according to Cantor, without any axioms. But Russell's paradox showed the need for something like axioms. > > > Where? �Why do you think taking a limit and taking cardinality should > > commute? �Should also the limit of te sequence of integral of functions > > be equal to the integral of the limit of a sequence of fuctions? > > If an infinite set exists as a limit, then it has gotten from the > finite to the infinite one by one element. During this process there > is no chance for any divergence between this set-function and its > cardinality. WM offers no proof that infinite sets can exist ONLY as limits, nor how they must be formulated if they are limits, so, as usual, the rest of his speculations are irrelevant.
From: Dik T. Winter on 10 Dec 2009 21:28
In article <89fb6e91-b6b1-4926-afca-820492e3cc6d(a)r24g2000yqd.googlegroups.com> WM <mueckenh(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. But apparently you are of the opinion that you are only allowed to define things that do exist. In that case it is better that you refuse to use proofs by contradiction. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |