Cantor Confusion
in [Math]
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From: mueckenh on 10 Feb 2007 05:57 On 9 Feb., 09:05, Virgil <vir...(a)comcast.net> wrote: > > Instead of a path P consider the set S of nodes K which belong to a > > path P. Do your calculation and arguing. Then substitute P for S to > > have a brief notation. > > One cannot determine merely from a set of nodes for a given tree whether > that set of nodes is or is not a path. > > One can. One has the coordinates of the node. Further there is no problem unless we consider special paths, because every node belongs to some paths. The consideration of special paths, however, can be restricted to one path, e.g., that one at the outmost left hand side. > > Absent verification that a set of nodes is the set of nodes in a path in > a tree, there is nothing about any such set of nodes that is not true of > any set of nodes.- Da stelle mer uns einfach janz dumm und retten so vleicht d set teori doch noch? You mean enough stupidity would be sufficient to safe set theory? You know that you cannot name most real numbers (= paths). But that does not hinder you to believe in the existence of these numbers, although you cannot even articulate any application of such a number. >> How can we show that a number without name does exist? > By showing that there are more numbers than names. An ordinal number is a name. It can be used to distinguish two entities. Ordinal numbers (not only natural numbers) are used in set theory as indexes, as you may learn when you study set theory. Your statement means showing that there are more names than names. Couldn't we generalitze it to showing that there are less names than names? > And this has been > done several times in these threads, despite WM's continuing ignorance. It has been shown that the number of names is not the number of names. Yes that 's an important theorem of set theory. And it is very typical for its usefulness as the foundation of mathematics. > When you can create enough names to name them all, then none of them > need be nameless. As every existing number is a name, nothing has to be created with existing numbers. Only the cherubim of your matheology do not fit in the problem. > In English "finished" conveys "there was", not "there is". "Finished infinity" means "infinity has been finished /abolished by finishing / completing it". (It is a play with words.) > What are the axioms and primitives of that system, as it does not seem > to be represented in English. If it is merely one of those Teutonic > things like the Gotterdamerung series, never mind. Yes, it is something like Goetterdaemmerung. >> Cantor's, Hilbert's, Fraenkel's etc. (Informal, of course.) > Which of them wrote it in English? Cantor (though he wrote a very bad English - at those times it was more important to learn Latin and Greek) Fraenkel published books in English. > As I recall one of Shubert's symphonies exists but is neither completed > or finished. Schuberts symphony no. VIII has been completed by several composers. The results are comparable to the completion of infinity by set theorists.: You better leave it as it was. > In analysis one regularly deals with infinite sequences and series which > are no more "completed" or "finished" than the set of naturals, The set of natural is the union of all finite segments. The binary tree is also the union of all finite trees. But the power set of the natural numbers is not the union of all finite sets of natural numbers because this would be a countable union. Do you see the problem? The union of finite trees and all the paths belonging to this union is countable. In order to get something uncountable you must starrt off with infinite sets. What part of N can this be? Every finite number is covered by a finite set. There must be something infinite which can never be covered by finite segments like {1,2,3,...n}. What could this be? Only finite numbers in N cannot not be responsible. Regards, WM
From: Franziska Neugebauer on 10 Feb 2007 06:13 mueckenh(a)rz.fh-augsburg.de wrote: > On 9 Feb., 09:05, Virgil <vir...(a)comcast.net> wrote: > >> > Instead of a path P consider the set S of nodes K which belong to a >> > path P. Do your calculation and arguing. Then substitute P for S to >> > have a brief notation. >> >> One cannot determine merely from a set of nodes for a given tree >> whether that set of nodes is or is not a path. > >> > One can. Then please do so: given tree T := { a, b, c, d, e, f, g } given set of nodes S := { a, b, c } Tell us whether S is a path in T. And please explain that. > One has the coordinates of the node. The what? F. N. -- xyz
From: mueckenh on 10 Feb 2007 08:21 On 9 Feb., 14:34, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > So a statement like "every element is a set > > > in ZFC" is false. > > > > It is used by most authors of text books on set theory. Talk to them > > about their errors. I am no interested in what they use to denote the > > elements (or sets) of their erroneous theory. > > But they give a proper foundation for that statement. So you see now that you were wrong? > You do not. You > pull that statement out of thin air. I did not support it because to support all familiar knowledge hre would lengthen these contributions overdue. > > > Read what I wrote. P(i) (uppercase P) is a set of paths, p(i) (lowercase > p) is a path. (This is from your notation from an earlier article.) I > state: > (1) p(0) U p(1) U p(2) U ... = p(oo) > (2) P(0) U P(1) U P(2) U ... != P(oo) > see the difference? The union of paths establishes the infinite path. > The union of *sets* of paths does not establish the set of infinite paths. The union of sets of paths P(0) U P(1) U P(2) U ... contains all finite the segments of the path p, p(0) U p(1) U p(2) U ... and, therefore, establishes p(oo). This is the same for every infinite path p, q, r, .... of the tree. P(0) U P(1) U P(2) U ... contains all finite the segments of the path q, q(0) U q(1) U q(2) U ... and, therefore, establishes q(oo). And so on: every infinite path r(oo) , s(oo), t(oo), ... of the tree is established. Therefore every infinite path of the set P(oo) is established by the union of the sets P(n): With every member, the set P(oo) is established by P(0) U P(1) U P(2) U ... More is not required. Regards, WM
From: mueckenh on 10 Feb 2007 08:29 On 9 Feb., 14:53, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <1171011850.731985.236...(a)a75g2000cwd.googlegroups.com> mueck....(a)rz.fh-augsburg.de writes: > > > On 8 Feb., 13:48, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > ... > > > > Why not. IIIII can be abbreviated by 5 or V or your proposal or .... > > > > > > Again, still no definition. So IIIIII can be abbreviated 5I, I5, VI or > > > IV? And IIIIIIIIII can be abbreviated 55 or VV? > > > > That is a matter of convention. Therefore the unary represantation, > > which is not a matter of convention, is preferable. > > Also unary representation is a matter of convention in my opinion. But it is requiring the least possible convention. III oo abc all that is understandable as 3 without other convention than that "all" is the start of this sentence and not another representation of 3. > > I believe that without any other cultural contact two persons could > > start communicating by unary numbers. > > Yes, your belief. Even my firm belief. > > > > > If you use "natürliche Zahl" or "N" does not matter. You use something > > > > not yet defined (and never defined by Peano). > > > > > > You apparently do not know how a recursive definition works. (1), (2) > > > (when properly corrected) define the natural numbers. (3) defines the > > > set of natural numbers. > > > > I know it. Cantor already used it (see p. 128 of my book). That does > > not mean that it is good. > > (1), (2) have not to be corrected (they appear in several text books). > > Your assertion that " x is in N" and "x is a natural number" were > > different is wrong. Both has to be defined. > > But, in (1) you state "is a natural number", in (2) you use "is in N", > without stating anywhere in that article that the two things mean the > same. So the combination of those two is *not* a proper definition. In the articles presented here one cannot state every self-evident convention. > > > > > N is only an abbrevation of set of natural numbers. > > > > > > An undefined abbreviation. When doing definitions you should do it the > > > correct way. See: > > > We are going to define natural numbers: > > > (1) 1 is a natural number > > > (2) if a is a natural number, the successor of a is also a natural > > > number > > > These two together define the natural numbers. > > > > Not yet. According to your definition also -7 and pi could be natural > > numbers. > > How do you come to that conclusion? At which step is either -7 or pi > generated by (2)? It is not excluded. > > > But we could write your two axioms also as: > > > We are going to define N: > > > (1) 1 is in N > > > (2) if a is in N, the successor of a is also in N. > > > These two together define N. > > Right. But not as you did: > (1) 1 is a natural number > (2) if a is in N, the successor of a is also in N. > These two together define the natural numbers. > By this definition only 1 is a natural number, because you can not > even start with statement (2). I copied these statements from my book without the explanation which is familiar to everyone discussing here. > > > > > Correct. There is no infinite line existing. All we assume is a line > > > > longer than any line we have measured yet. > > > > > > Such lines also do not have physical existence. Each and every physical > > > line has a width smaller than anything measured yet. And I do not think > > > that physical lines are really straight either. > > > > The physical existence of a line is "a measurable distance" between > > two points. The points exists as sets of coordinates. > > Oh. What is a circle? The set of points with a fixed distance from a given point. > What is a parabola? An invention by Archimedes. I am sure you know the definition in terms of distances. Regards, WM
From: mueckenh on 10 Feb 2007 08:33
On 9 Feb., 16:22, "William Hughes" <wpihug...(a)hotmail.com> wrote: > On Feb 9, 4:15 am, mueck...(a)rz.fh-augsburg.de wrote: > And at this point you acknowlege that is it possible > to define the sparrow of E. So the sparrow > of E exists. Do you think that everything exists which I acknowledge? > If we decide to call the sparrow of E a number, then it > is not a natural number But it does not mean that this sparrow is alive. If we decide to call a number between 1 and 2 a natural number, then this is a wrong definition. > and this statement is not > a contradiction. It is. > > No statement you make about things that are true > of every set with finite cardinality, or things that > are true for every natural number, can be used > to show something about the sparrow of E. > The set E is not a set with finite cardinality > and the sparrow of E is not a natural number. > The fact that E is composed of sets with finite > cardinality does not mean that E is a set > with finite cardinality. The finite cardinality has been proved by complete induction. > The fact that the cardinality > of E can be seen as the limit of natural number > does not mean that the cardinality of E must > have the same properties as the natural > numbers. A limit of a sequence does not > have to have the same properties as the elements > of a sequence. A limit of a sequence (a_n) has to have the Cauchy property. omega - n = omega does not satisfy it. > > > > > > Extending the concept of cardinality to include > > > potentially infinite sets does not lead to > > > a contradiction. > > > Unless you say that it is a number larger than any natural number. > > No. If you extend the concept of cardinality to potentially > infinite sets, then the cardinality of a potentially infinite set > is not a natural number, It is not a number and cannot be a number. It can only be the property that the natural number which is the cardinality of the set in present state can grow. But the property "can grow" is as little a number as the property can eat hot dogs or can drive a red car is a number, let alone an infiite number. Regards, WM |