In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 17 Jun., 21:59, MoeBlee <jazzm...@hotmail.com> wrote: > > > > I never said a real number can be defined by an infinite sequence. > > All finite words belong to a countable set. If you exclude infinite > words (sequences) then there is no chance for uncountability. > > > > > A real number > > > can be defined only by a finite word. But there is no diagonalization > > > over finite words. > > > > Without even commenting on what you mean or whether it is true, it > > does not refute that the formalized argument is first order logic > > applied to axioms and incontrovertible > > Incontrovertible is religion. Because its adherents exclude > refutations from their perception.
Something being "Incontrovertible" in FOL means you can't controvert it that while playing by the rules of FOL.
AS far as I am aware, there are no definite set of rules of logic like FOL (first order logic) for religions. > > > You've not said what "wrong" assumption I've "started with". > > The possibility of an infinite sequence of infinite sequences that can > be completed in order to obtain a completed "anti-diagonal" sequence.
That possibility is a consequence of an axiom set like FOL plus ZFC. And what is possible within such an axiom set in not constrained by whatever other axioms WM wishes to impose, nor even by WM's views on "reality". > > > All of this business of yours does not refute what is simply > > introvertible, that a formal proof exists in the manner I've > > mentioned. > > There may be a proof. But as the result is wrong the proof is not > worth much.
Since it has not been proven wrong in FOL + ZFC, or whatever other system it was proved in, the proof stands.
There is a form of pure mathematics which operates much like games, in that one sets rules and then plays within those rules. For this sort of mathematics, those like WM who insist on repeatedly breaking those rules are viewed as cheaters, and deserve to be. > > > > > The translation of these notions into your "incontrovertible" theory > > > is the weak point. > > > > NO, you did not listen to what I said. I did NOT say anything about an > > incontrovertible THEORY. Rather, I said it is incontrovertible that a > > certain finite sequence of finite sequences of symbols exists. > > But this finite sequence leads to the result that an uncountably > infinite sequence of infinite sequences exists. And that is wrong.
WM's assertion of error conclusion is cheating. > > > > > > > Does ZFC not prove that all constructible numbers are countable? > > > > I don't know. What is the definition IN to set the rules for otherTHE LANGUAGE of ZFC of > > 'constructible number'? > > > > Anyway, I have no idea how you think that bears upon what I just > > wrote. > > So there seems to be a gap in ZFC. But it is easy to prove that in > fact there are only countably many constructible numbers.
WM is free to set whatever rules he wants for his games, but is not free to override the rules that others have set for their games.
And that form of "cheating" is exactly what he is forever trying to do.