Cantor Confusion
in [Math]
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From: mueckenh on 20 Feb 2007 16:36 On 20 Feb., 16:11, "William Hughes" <wpihug...(a)hotmail.com> wrote: > You wrote (then snipped) > > M: [t]he property that every set of even natural numbers must contain > numbers > M: larger than its cardinal number, is correct, unless the set > contains > M: unnatural numbers. > > As I noted this is false even in Wolkenmueckenheim. You may note what you want. The above statement is true if the statement is true that even in an infinite chain we can conclude from a<b<c<...<z that a is not larger than z. (Should you continue with personal insults, I will stop this discussion with you.) Regards, WM
From: mueckenh on 20 Feb 2007 16:40 On 20 Feb., 16:18, "William Hughes" <wpihug...(a)hotmail.com> wrote: > On Feb 20, 8:56 am, mueck...(a)rz.fh-augsburg.de wrote: > > > > > > > On 19 Feb., 14:53, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > > In article <1171889781.807587.262...(a)s48g2000cws.googlegroups.com> mueck...(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. > > > like infinity. But it is impossible that both follows, infinity and > > naturality. > > Since no one has claimed that infinity is a natural number > this is not a problem. (Note the claim that "infinity is > the union of natural numbers" and "infinity is a natural > number" are two different claims.) Note that the number of natural numbers can be mapped one-to-one on the natural numbers: 1,2,3,...,n <---> n It is easy to see that natural nunmbers alone do not bring together an infinite number of numbers. Regards, WM
From: Virgil on 20 Feb 2007 16:45 In article <1171981466.237613.54010(a)v33g2000cwv.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Every path of the tree is a special subset. (Not every subset is > a path.) But there are only countably many finite subsets and > countably many sets of subsets which belong to one and the same path. WRONG! There are uncountably many subsets of every infinite path, though only countably many of them are paths in some initial finite tree.
From: mueckenh on 20 Feb 2007 16:45 On 20 Feb., 20:37, "MoeBlee" <jazzm...(a)hotmail.com> wrote: > On Feb 17, 12:52 am, mueck...(a)rz.fh-augsburg.de wrote: > > > 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. > > Natural numbers are counting the elements of natural sets, i.e., of > > sets which exist in reality (in nature, as Cantor woud have said). > > So if we call 0 and the positive whole numbers 'mamtural numbers' It is enough to call them ordinal numbers. It will not change mathematics but will help to avoid confusion about the term natural number. When I say that every set of even natural numbers contains numbers larger than its cardinality, then I am right with my favourite meaning of natural numbers but wrong with Bourbaki's. Regards, WM
From: MoeBlee on 20 Feb 2007 16:52
On Feb 20, 1:40 pm, mueck...(a)rz.fh-augsburg.de wrote: > natural nunmbers alone do not bring together an > infinite number of numbers. Right. In Z set theoreis, without the axiom of infinity, we can define 'is a natural number' and prove the existence of any natural number, but we can't prove the existence of a set that has as members an infinite number of natural numbers. So we adopt the axiom of infinity. MoeBlee |