constructible = unconstructible
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From: WM on 10 Jun 2010 09:30 On 10 Jun., 14:10, William Hughes <wpihug...(a)hotmail.com> wrote: > On Jun 10, 7:34 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > <snip> > > > if we divide 1 by 9, then 0.111... does exist according > > to the above definition. > > If we write it in a single line, nobody will disagree. > > If we write it such that every next 1 is placed in a new line, like > > > 0.1 > > 0.11 > > 0.111 > > ... > > > then everybody will disagree that all 1 are in one line but all 1 > > "must be there somehow". > > Yes and outside Wolkenmuekenheim, everyone is correct. > However inside Wolenmuekenheim there is a 1 in 0.111... > that is not in the list. What is flaming good for? My proof uses complete induction. One can believe in it or not. But I believe that it is valid for every line of the list. Therefore I do not believe that your arguing "all 1's are in the list but not all 1's are in the same line" is wrong, namely disproved by induction. And if you think in logocal terms and understand that induction holds for every natural number, here every line, then you must agree. Sometimes people say that induction does not hold for the complete set N. But that is not at all the question here. Here we have to prove that every line contains all 1's of its predecessors. > > > This leads to the easily disprovable claim > > that there are at least two lines required to contain all 1's. > > Nope, it leads to the easily provable claim that a > set of lines, S, that contains all nodes must > contain an infinite number of lines. Nonsense. If someone claims that infinitely many are necessary, instead a single one, then he should know at least two of them. > Since any set > of infinite lines is sufficient, and there exist disjoint > sets of infinite lines, it follows immediately that > no two specific lines are required. It follows immediately by induction, that no two lines are required. Please excuse if I trust in complete induction more than in your shaky assertions. Regards, WM
From: WM on 10 Jun 2010 09:33 On 10 Jun., 14:17, William Hughes <wpihug...(a)hotmail.com> wrote: > On Jun 9, 10:37 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > On 9 Jun., 15:19, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > It is well known that the set of finite descriptions > > > is countable (though not computable). > > > The set of all names which unavoidably include all nunbers is > > computable. > > The set of all names, S is computable. I showed you how it goes. > The set of all computable numbers, R is a subset of S. > A subset of a computable set may or may not be computable. > R is not computable. You may be right. But who is interested in that possible fact? A subset can never have larger cardinality than the set. And the set is countable. Regards, WM
From: William Hughes on 10 Jun 2010 10:00 On Jun 10, 10:33 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > On 10 Jun., 14:17, William Hughes <wpihug...(a)hotmail.com> wrote: > > > On Jun 9, 10:37 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > On 9 Jun., 15:19, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > It is well known that the set of finite descriptions > > > > is countable (though not computable). > > > > The set of all names which unavoidably include all nunbers is > > > computable. > > > The set of all names, S is computable. > > I showed you how it goes. > > > The set of all computable numbers, R is a subset of S. > > A subset of a computable set may or may not be computable. > > R is not computable. > > You may be right. But who is interested in that possible fact? A > subset can never have larger cardinality than the set. And the set is > countable. > It is well known that the set of computable numbers is countable (though not computable). - William Hughes
From: WM on 10 Jun 2010 10:32 On 10 Jun., 16:00, William Hughes <wpihug...(a)hotmail.com> wrote: > On Jun 10, 10:33 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > > > On 10 Jun., 14:17, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > On Jun 9, 10:37 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > On 9 Jun., 15:19, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > > It is well known that the set of finite descriptions > > > > > is countable (though not computable). > > > > > The set of all names which unavoidably include all numbers is > > > > computable. > > > > The set of all names, S is computable. > It is well known that the set of computable numbers > is countable (though not computable). And what is the real part or numerical value of an uncomputable real number? Regards, WM
From: William Hughes on 10 Jun 2010 10:41
On Jun 10, 11:32 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > On 10 Jun., 16:00, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > On Jun 10, 10:33 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > On 10 Jun., 14:17, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > On Jun 9, 10:37 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > > On 9 Jun., 15:19, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > > > It is well known that the set of finite descriptions > > > > > > is countable (though not computable). > > > > > > The set of all names which unavoidably include all numbers is > > > > > computable. > > > > > The set of all names, S is computable. > > It is well known that the set of computable numbers > > is countable (though not computable). > <important context restored> The set of all computable numbers, R is a subset of S. A subset of a computable set may or may not be computable. R is not computable. > And what is the real part or numerical value of an uncomputable real > number? Something that cannot be computed. If you take the position that something that cannot be computed does not exist then neither an uncomputable number nor a list of all computable numbers exist. - William Hughes |