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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/
From: Dik T. Winter on 10 Dec 2009 21:50 In article <fec95b83-39c5-4537-8cf7-b426b1779f84(a)k17g2000yqh.googlegroups.com> WM <mueckenh(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? > > 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. > > > 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. Within ZF a sequence is an ordered set. But as you refuse to distinguish beteen an ordered set and a non-ordered set, I think this goes beyond you. > > > 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. > > > 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? > > 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. In what way do numbers like 1/3 exist in your tree? Not as a path, apparently, but as something else. Similar for 'pi' and 'e'. 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. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 10 Dec 2009 21:40
In article <5333fb9a-1670-4fcc-85d3-25e75fb5bd1d(a)f16g2000yqm.googlegroups.com> WM <mueckenh(a)rz.fh-augsburg.de> writes: > 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. I have never seen there a proper definition of omega. > > There are no concepts of mathematics without definitions. > > So? What is a set? Something that satisfies the axioms of ZF for instance. > > > 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. Yes, so what? You are arguing against current set theory, in the time of Cantor it was still being developed. > > 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. If a function exists as a limit, then it has gotten from the finite to the infinite one by one step. During this process there is no chance of any divergence between the function and the integral. Now, what is wrong with that reasoning? Stronger: lim(n -> oo) 1/n = 0 1/n > 0 If a number exists as a limit, then it has gotten from the finite to the infinite one by one step. During this process there is no chance of any divergence between the element and the inequality. What is wrong with that reasoning? You are assuming that taking a limit is a final step in a sequence of steps. In the definition I gave for the limit of a sequence of sets there is no final step. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |