On 6 Apr., 21:54, Virgil <vir...@ligriv.com> wrote:
> > You were right. The reason is: There are at most countably many finite > > definitions like e = SUM1/n!. That is undisputed. So if there should > > be uncountably many reals, most of them cannot be defined - or can > > only be defined by infinite sequences. > > Pi and sqrt(2) are both reals defined other than by infinite sequences.
Of course. They exist as elements of the countable set of finite names. > > > But that means the same as > > being undefined, because none of those sequences defines a number > > unless you know the last digit - which is impossible. > > Thus WM here claims that neither pi nor sqrt(2) can be defined!
> Now WM claims that neither pi nor sqrt(2) cannot be used in math because > they cannot be communicated.
You err. > > And here comes a > > > simple proof that the notion of uncountablility is in fact nonsense: > > Construct all real numbers of the unit interval as infinite paths of > > the complete infinite Binary Tree. It contains all real numbers > > between 0 and 1 as infinite paths i. e. infinite sequences of bits. > > [...] > > The complete tree contains all infinite paths. The structure of the > > Binary Tree excludes that are any two initial segments, B_k and B_(k > > +1), which differ by more than one infinite path. > > Actually, there are infinitely many, uncountably many,
> > (In fact no B_k does > > contain any infinite path > > But each , as a FISON, finite initial sequence of nodes, is contained
You err. B_1 contains 0 and 0.0. B_2 contains 0 and 0.0 and 0.1.
> > Hence either there are only countably many infinite paths. > > Or uncountably many infinite paths come into the tree after all finite > > steps of the sequence have been done. > > In this case
again you err. >
> WM is great as assuming things, but abominable at proving things,
You err. It looks so to you, because you cannot even understand the most simple proofs like this one:
Cantor's argues that the diagonal will differ from each entry at some digit n. I have proved that this is false, since for every n there are infinitely many entries with the same FIS d_1, ..., d_n as the anti- diagonal. Since this is true for every n, it cannot be false for any n.