Cantor Confusion
in [Math]
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From: William Hughes on 20 Feb 2007 10:18 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.) - William Hughes
From: Randy Poe on 20 Feb 2007 10:28 On Feb 20, 8:47 am, mueck...(a)rz.fh-augsburg.de wrote: > On 19 Feb., 14:36, "William Hughes" <wpihug...(a)hotmail.com> wrote: > > > > > 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. > > They can be used in certain cases. But we have no common basis for > discussion. > You believe that if the chain > a < b < c < d <... < z is only infinitely long, > then a = z is possible. What true statement is this (false) parody based on? - Randy
From: MoeBlee on 20 Feb 2007 14:37 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' instead of 'natural numbers', that should be okay. And the mathematics won't be a bit different except for our having to cross out 'natural number' and replace with 'maturual number'. MoeBlee
From: MoeBlee on 20 Feb 2007 15:12 On Feb 18, 1:29 am, mueck...(a)rz.fh-augsburg.de wrote: > 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? Sometimes people do use terms like 'the 0th'. This ia a matter of the natural language used, and is not a matter of any importance for the formal mathematics. MoeBlee
From: cbrown on 20 Feb 2007 15:46
On Feb 20, 7:11 am, "William Hughes" <wpihug...(a)hotmail.com> wrote: > On Feb 20, 8:47 am, mueck...(a)rz.fh-augsburg.de wrote: > > > > > On 19 Feb., 14:36, "William Hughes" <wpihug...(a)hotmail.com> wrote: > > > > 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. > > > They can be used in certain cases. > > Not alone. i and ii are not enough to > show iii. When someone says > "iii is false" you reply "but i and ii > are true". Since i and ii are not enough > to show iii your replies are empty. > > > > 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. > E is a counterexample. > My question remains. How many times are you going to > rephrase this and get it wrong? > Umm, floor of e^(10^88)? Cheers - Chas |