In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> On 24 Jun., 11:04, Virgil <Vir...@home.esc> wrote: > > > > Thise prodedure is abbreviated by > > > (..., An, ... A2, A1, A0, L0) > > > but of course there cannot appear at any stage a list without first > > > line. > > > > But that list can be rearranged into a more standard form, for example > > with the An and the members of L0 listed alternatingly, and from such a > > rearrangement a non-member anti-diagonal can be constructed. > > What should that be good for? The countable set under investigation > contains the antidiagonal of the arragement that I prescribed.
The "arrangement" you prescribe does not suport antidiagonal construction without rearrangement into standard list form.
You claim a function from N to R (the reals) or B (the set of endless binary sequences) which contains its own antidiagonal. Nonsense! Cantor proved no such containment possible, and nothng WM has done has shown that proof flawed. > > Your arguing reminds of another argument: I can eat cherries, > therefore the list is neither complete nor incomplete.
That certainly reminds me of many of WM's arguments. > > I for my person would not accept that. Perhaps set theorists have > another taste.
Mathematicians accept that in certain set theories there re uncountable sets. > > > > > > It is as simple as that. > > > > > It is not a big deal. Set theory shows many contradictions arising > > > from the idea, that from "every n in N can be constructed" it is > > > implied that "N can be constructed".
> > In, for example, FOL+ZFC, there is no assumption that every n in N "can > > be constructed". What is assumed is that there exists a set, along with > > all its elements, having the properties that we want for N and its > > elements , which we then call N. > > And being ready to be used for the construction of a bijection.
Such bijections from to suitable other sets can often be "constructed", though the more common term is "defined". > > > > This is wrong, because for every > > > > > n, there is an infinite set of unconstructed elements of N. > > > > But we do not "construct" any of them. Though we do construct various > > naming conventions for elements of N. > > Bijections from M to N are constructed. Sequences (having natural > indices) are constructed. Many constructivists do not believe in > uncountability but in N because N can be constructed.
If your "constructed" means "defined" then yes.
But in the Cantor theorem, the "antidiagonal" is defined (but not necessarily constructed) and since it defines a binary sequence not listed in the list from which it is defined, it shows that no list can include all such binary sequences, QED.