Cantor Confusion
in [Math]
|
From: Dik T. Winter on 18 Feb 2007 19:02 In article <1171789749.778902.224030(a)h3g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > On 16 Feb., 16:36, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: .... > > > > In none of them I do immediately see the number three. What I see > > > > is that there are three of something, not the number three. > > > > > > What is common to all? > > > > I stated that already just above. > > Yes. You said "three of something". That is exactly the meaning of the > number three. 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? -- 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 18 Feb 2007 19:35 In article <1171816407.391676.216390(a)v33g2000cwv.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: .... > The limit {1,2,3,...} = N does exist according to set theory. Why > should the limit {1}, {1,2}, {1,2,3}, ... = N not exist? There you go again. Set theory does *not* use limits. I know of *no* definition that gives the limit of a sequence of sets. So pray write down here how *you* define the limit of a sequence of sets. -- 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 18 Feb 2007 19:31 In article <1171790940.416072.43950(a)k78g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > On 18 Feb., 02:51, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article <1171702351.590890.177...(a)h3g2000cwc.googlegroups.com> mueck..= > .@rz.fh-augsburg.de writes: > > > On 16 Feb., 15:58, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > > > This is not a representation of 3 other than in a perverted > > > > > system, which calls 0 the first number, 1 the second and so > > > > > on. Of course {{{{}}}}, or better and easier {{{{, denotes the > > > > > fourth number which is 4 and not 3. > > > > > > > > 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? > > > > > > 0 may be the first (or better the zeroest) ordinal or cardinal number > > > (if you wish to have the empty set in the theory). Nevertheless it is > > > not the first natural number and not a natural number at all. > > > > But you said "perverted system which calls 0 the first number". The only > > place where it is called the first *natural* number is in Bourbaki and its > > 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. A human being in its twentieth year has the age 19. What is so strange about doing the same with ordinal numbers? The twentieth century encompasses the years 1901-2000. There are all kinds of places where the linguistic ordinal number does not match the mathematical ordinal number. (Note that the above holds for Western cultures, there are cultures where different conventions do apply. Like those cultures where the first year of a ruler starts a year after taking the throne, or that state where year 35 of the rule is actually the 29-th year.) > > But following your reasoning, {{{}}} is the third number, which is 3. > > BTW, I can quote you as saying: > > > > > > {{{}}} > > > > > It was page 93 of my book. > > > > I have seen it earlier than that. > > > By the way, above is only number 2 given. > > So earlier you said it is 2. What is it? > > 2 is the number called in set theory which I referred to here and in > chapter 7 of my book (I hope without too many errors). > 3 is the number called in any reasonalbe system. So you are trying to obfuscate the issue. > > If my house contains no dogs, in what way does the set of dogs in my house > > not exist in reality? > > There is no dog and no set of dogs, because it would be the same set > of cats. Your dog would be your cat. That is impossible. They would > not live together, let alone form the same (identical) set. In my house the set of dogs is the same as the set of cats. And indeed, my dog is my cat. But that is all pretty unmathematical. > > 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. > > > 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. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Michael Press on 18 Feb 2007 20:56 In article <1171790147.558130.297920(a)j27g2000cwj.googlegroups.com> , mueckenh(a)rz.fh-augsburg.de wrote: > On 17 Feb., 03:05, Michael Press <rub...(a)pacbell.net> wrote: > > In article > > > > <1171468278.284645.273...(a)v33g2000cwv.googlegroups.com>, mueck...(a)rz.fh-augsburg.de wrote: > > > On 13 Feb., 21:17, Virgil <vir...(a)comcast.net> wrote: > > > > In article <1171364856.226197.135...(a)l53g2000cwa.googlegroups.com>, > > > > mueck...(a)rz.fh-augsburg.de wrote: > > > > [...] > > > > > > > What about all existing sets with 3 objects, i.e., the fundamenal set > > > > > of 3? > > > > > > What fundamental set of 3 does WM refer to? > > > > > Olease read before writing. The set of all existing sets with 3 > > > objects. > > > > How many sets with three elements are there? > > No idea. Less than 10^100. But that is unimportant. Important is that > there is at least one set with 3 elements. It is important to me. How many? Show me a set with three elements. -- Michael Press
From: mueckenh on 19 Feb 2007 07:56
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. > > 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. 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. 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. Regards, WM |