Cantor Confusion
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From: Virgil on 20 Oct 2006 19:37 In article <1161378001.475899.279610(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > > Every diagonal number is constructed and, therefore, is constructible. > > > > When constructed from a constructable list. > > What is an unconstructable list? Do you call any undefinite mess a > list? > Any list is construtced. Any diagonal number is constructed. A list of reals for the Cantor construction need not be made up of constructable numbers, it is only required that the nth number be constructable to the nth decimal place.
From: William Hughes on 20 Oct 2006 19:46 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueck...(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > > A constructible number is a number which can be constructed. Definition > > > > > obtained from Fraenkel, Abraham A., Levy, Azriel: "Abstract Set > > > > > Theory" (1976), p. 54: "Why, then, the restriction to the digits 1 and > > > > > 2 in our proof? Just to kill the prejudice, found in some treatments of > > > > > the proof, as if the method were purely existential, i.e. as if the > > > > > proof, while showing that there exist decimals belonging to C but not > > > > > to C0, did not allow to construct such decimals." > > > > > > > > > > Definition (by me): A number which can be constructed like pi, sqrt(2) > > > > > or the diagonal of a list is that what I call constructible. If you > > > > > dislike that name, you may call these numbers oomflyties. Anyhow that > > > > > set is countable. > > > > > > > > Nope. By the definitions you use, that set is not countable. > > > > > > > Every set of constructions is countable due to the finite alphabet of > > > any language. > > > > > > No. If you restrict yourself to computable functions you have some > > counterintuitive results. Assume that > > the language you are working in has a finite alphabet. Then the set > > of all finite strings in the language is listable using a computable > > function > > (use dictionary order). And so the set of all finite strings is > > countable. > > Now, A, the set of all strings which define a computable number is a > > subset of the set of all finite strings. So A is countable, right? > > Wrong! > > It is not true that every subset of a countable subset is countable. > > It is true that every set can be well ordered and that any two sets can > be compared. Both are equivalent or one is equivalent to a sequence > (ordered subset) of the other. What you say is not counter intuitive > but wrong. > This theorem requires the use of arbitrary functions. If you want to say that the computable reals are countable then you need non-computable functions and there is a non-computable list of the computable reals. However, you assume that only computable functions (and hence computable lists) exist. Then: It is not true that there is a computable function which will list the elements of A (to do this you would have to be able to identify the elements of A in a computable manner, and to do this you have to solve the halting problem). - William Hughes
From: Virgil on 20 Oct 2006 19:47 In article <1161378187.155995.290420(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1161276574.792436.186750(a)i3g2000cwc.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > > > > Dik T. Winter schrieb: > > > > The reason that continuity plays no role is because the function of > > > > the > > > > number of balls in the vase when written as a function of t is > > > > discontinuous at infinitely many positions. > > > > > > But it is stepwise continuous. > > > > Yes, but it is infinitely many times not continuous in each neighbourhood > > of the limit point. > > Let t be the ordinal number of transactions. > > It *is* continuous like a staircase. Only if a ball can be put into the vase or removed a little bit at a time in a time interval of zero length, instead of all at once. Otherwise the staircase "riser" does not exist, and the graph has a discontinuity. > > > Translate balls as numbers and vase as set variable. More is not > > > required. > > > > That is still not a mathematical formulation. More is required. You > > need to actually state what you mean with 'number of balls in the vase > > at noon' or 'natural numbers in the set at noon'. > > The number of transactions t is then t = omega at noon. Actually, the number of transactions AT noon is 0, but there are lots of them in every neighborhood of noon. > > Also there is a > > time dependency that is not clearly stated. > > Time dependency can be eliminated if you count the transactions by > natural numbers t as I do. But it cannot be eliminated without violating the constraints of the experiment. > > > > The limit can in mathematics *always* be determined by the terms (if there > > is a limit). But the limit in no case defines the function value at the > > limit point. > > Then the irrational numbers as limit points are undefined. They ARE defined as LUBs or GLBs of suitable sets of rational numbers. But that is quite a different problem.
From: Dik T. Winter on 20 Oct 2006 20:21 In article <1161377915.999210.39660(a)m7g2000cwm.googlegroups.com> "MoeBlee" <jazzmobe(a)hotmail.com> writes: > mueck...(a)rz.fh-augsburg.de wrote: .... > Okay, now that I asked for a definition of the relation you mentioned, > you're not giving that definition, but instead giving a > combinatorical/numerical argument with more terminology. What is a > "load of edges"? What is the definition of "a path carries a load of > edges"? If this is standard terminology in graph theory, then please > forgive my ignorance and supply me with the standard definition. If it > is not standard terminology, then please give me your own definition. By this time you ought to know that Mueckenheim *never* gives definitions. Or actually states that he is not able to give a definition for a particular term. Asking for definitions from Mueckenheim is as useful as talking to an eel. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Virgil on 20 Oct 2006 21:17
In article <J7GMC6.LuH(a)cwi.nl>, "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote: > In article <1161377915.999210.39660(a)m7g2000cwm.googlegroups.com> "MoeBlee" > <jazzmobe(a)hotmail.com> writes: > > mueck...(a)rz.fh-augsburg.de wrote: > ... > > Okay, now that I asked for a definition of the relation you mentioned, > > you're not giving that definition, but instead giving a > > combinatorical/numerical argument with more terminology. What is a > > "load of edges"? What is the definition of "a path carries a load of > > edges"? If this is standard terminology in graph theory, then please > > forgive my ignorance and supply me with the standard definition. If it > > is not standard terminology, then please give me your own definition. > > By this time you ought to know that Mueckenheim *never* gives definitions. > Or actually states that he is not able to give a definition for a > particular term. Asking for definitions from Mueckenheim is as useful as > talking to an eel. And there are further resemblances. |