Cantor Confusion
in [Math]
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From: William Hughes on 19 Feb 2007 08:36 On Feb 19, 7:56 am, mueck...(a)rz.fh-augsburg.de wrote: > On 18 Feb., 15:53, "William Hughes" <wpihug...(a)hotmail.com> wrote: > > > > You are right. The claim in its generality is clearly wrong, > > > So stop using it. Stop claiming > > > This holds for every initial finite segment therefore > > it holds for the set. > > No. Then we must also stop claiming that the set which is the union of > all initial segments {1,2,3,...,n} contains only natural numbers. No. The statement This holds for every initial finite segment therefore it holds for the set. is not true, even if the set contains only natural numbers. Take a property X. Take a potentially infinite set A (say the union of all initial segments {1,2,3,...,n}). Then, as you note ("The claim in its generality is clearly wrong") the statements: i: Every initial segment {1,2,3,...,n} has property X ii: Every element of A that can be shown to exist is a natural number Do not imply iii: A has property X. Sometimes i and ii are true and iii is true. Sometimes i and ii are true and iii is false. Statements i and ii cannot be used to prove iii. > > > > > If you want to prove something holds for the set you > > cannot use induction. > > That depends on the question. > > > Induction can only show that something > > holds for every initial segment. As you have now noted, > > this says nothing one way or the other about the set. > > Not every property obeys induction. For instance, the property of > having an even number of elements does not obey induction. However, the property of "being either odd or even" does. The property of "being either prime of composite" does. The property of having a fixed largest element does. The property of having a cardinality does. E is not either odd or even, E is not prime of composite, E does not have a fixed largest element, E does not have a cardinality. Whether or not you can use induction to show that every initial segement has property X, you cannot use induction to show that the union of every initial segment has property X. > But the > property that every set of even natural numbers must contain numbers > larger than its cardinal number, is correct unless the set contains > unnatural numbers. No The statement is not true for E E does not contain unnatural numbers. How many times are you going to rephrase this and get it wrong? Hint: Next time check for yourself if E is a counterexample. > As long as we have a set S c N of the form > {2,4,6,...,2n,...} > then some elements of S are larger than card S, if card S is in > trichotomy with natural numbers. Let A be a potentially infinite set. Then card A does not exist (in Wolkenmueckenheim,). A set of the form {2,4,6,...,2n,...} is potentially infinite [there is only one such set, E]. So your statement reduces to the trivial Sets of the form S_n={2,4,6,...,2n} contain an element larger than card S. - William Hughes
From: Dik T. Winter on 19 Feb 2007 08:53 In article <1171889781.807587.262850(a)s48g2000cws.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > On 18 Feb., 15:53, "William Hughes" <wpihug...(a)hotmail.com> wrote: > > > > You are right. The claim in its generality is clearly wrong, > > > > So stop using it. Stop claiming > > > > This holds for every initial finite segment therefore > > it holds for the set. > > No. Then we must also stop claiming that the set which is the union of > all initial segments {1,2,3,...,n} contains only natural numbers. That is not proven using induction. It follows from the definition of the union. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 19 Feb 2007 09:05 In article <1171890114.237911.143650(a)k78g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > On 19 Feb., 00:57, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article <1171789678.561730.262...(a)t69g2000cwt.googlegroups.com> mueck...(a)rz.fh-augsburg.de > > > The union of sets of alle finite paths is the union of all finite > > > trees, and this union is the complete tree T(oo) if the union of all > > > initial segmens of N is N. > > > > Wrong. The union of sets of paths is *not* the tree. > > The union of sets of paths is one set of paths. The union of this set > of paths is a set of nodes. Yes. So your statement: "The union of sets of all finite paths is the union of all finite trees" is false. Correct would be: "The union of the union of sets of all finite paths is the union of all finite trees". > > There is a huge > > difference between the union of sets of paths (which is a set of paths) > > and the union of paths (which is a set of nodes). > > Of course there is a difference. But the tree is the union of the set > of paths. Yes, but it is *not* (as you stated) the union of *sets* of paths. > > > This is the fundamental problem of set theory. The union of all finite > > > segments of N is and must be the same as the union of all natural > > > numbers. But while the latter union is infinite by defintion and by > > > axiom, the union of all finite segments cannot be the infinite segment > > > N (because N is not contained in this union). > > > > It need not be and is not in that union. That is your funamental > > misunderstanding. It *is* that union. Moreover, the union of all finite > > numbers (using von Neumann representation) *is* N, it does not contain N. > > The union of segments {1}, {1,2}, {1,2,3}, ... of the set N of natural > numbers does not contain N = {1,2,3,...}. But if all these segments > are subsets of a set R then also N is a subset of this set R. > Now replace R by T. When I state "is in" I mean is an element of, not is a subset of. Trivially each set is a subset of itself. And so I also do interprete "does contain". But if you mean with "does contain": "is a subset of", N does contain N. But N is *not* an element of N. > > > But the infinite union of finite segments is obviously incapable of > > > yielding an infinite path, as you say above: "there can be no infinite > > > path in that union because none of the constituent sets contains an > > > infinite path." > > > > There is not infinite path as an element *in* that union, that union *is* > > the infinite path. Pray look at the difference. > > And this path belongs to the tree. Strawman. > > But that one is correct. But in P(0) U P(1) U P(2) U ... you are *not* > > uniting path, you are uniting sets of paths. > > Yes. You get a set of paths. The paths of this set can be united to > get a set of nodes, namely the tree. > > > T(2) = {0, 1, 2, 3, 4, 5, 6} > > so T(2) != P(0) U P(1) U P(2). > > T(2) = U(P(0) U P(1) U P(2)). And so your statement T(2) = P(0) U P(1) U P(2) was incorrect. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 19 Feb 2007 09:06 In article <1171890209.831371.70880(a)h3g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > On 19 Feb., 01:02, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: .... > > Not in my opinion, and I think not in mathematics. If the meaning of > > 3 is "three of something" how than do we calculate "three of something" > > times "three of something"? Or "nine of something" divided by > > "three of something"? Or more concrete, if I divide nine apples by > > three apples what is the result? > > three of something, where the "something" here means the unit. What unit? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 19 Feb 2007 09:19
In article <1171890516.474583.210020(a)p10g2000cwp.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > On 19 Feb., 01:31, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: .... > > > > > > Can you tell me a form of set theory where 0 is *not* the first > > > > > > ordinal or cardinal number? If so, how many elements does the > > > > > > empty set have in such a system? .... > > > In all set theory 0 is called the first ordinal number, but in fact it > > > is the zeroth one. Why do you start counting ordinals with 0 but start > > > counting ordinally with 1? > > > > So your "natural number" above was a red herring? A human being in its > > first year has the age 0. > > Before completing his first year, the being has the number of years > which comes before 1. > The first number drawn in lottery may be the 7. > > The first is that ordinal number which we start with. So your statement above: "In all set theory 0 is called the first ordinal number, but in fact it is the zeroth one" was nonsense? > It is the first > ordinal number. 0 is not commonly accepted as an ordinal number. That > is why usually we speak of first, even set theorists do so. But > however this may be, 0 is not a natural number. Well Bourbaki disagrees with you. But can you tell me which current mathematician does not accept 0 as an ordinal number? > > > > If you think {} to be an unnatural set, so be it (that is not mathematics, > > > > because there is no mathematical definition of natural set). > > > > > > There not even a mathematical definition of an unnatural set. > > > > Indeed. There is only a definition of set. > > Not even that. <http://en.wikipedia.org/wiki/Set> > > > > But if > > > > somebody asks how many coins I have in my purse, he is asking for the > > > > cardinality of the number of coins in my purse. And I can correctly > > > > answer 0 at some times. In that case the set is the set of coins in > > > > my purse, and that can be empty (and is quite often in reality). > > > > According to current definitions and axioms in set theory, {} *is* a > > > > set. > > > > > > That is known to me but will not prevent me from criticising it. > > > > Well, I do know you do not like the axiom of infinity, but the other > > axioms of ZF imply the existence (and uniqueness) of the empty set. > > Of course. There is usually the axiom of its existence. Depends on the formulation of the axioms. The axiom set of Kunen does *not* contain the axiom of the empty set. Existence and uniqueness follows from the other axioms (extensionality, regularity/foundation, specification/separation/restricted comprehension, pairing, union, replacement, infinity and power set).. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |