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.
> 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.
> 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. > > > 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. > > > > Does ZFC not prove that all constructible numbers are countable? > > I don't know. What is the definition IN THE 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. > > > Is Cantor's diagonal not a constructible number? > > This reveals your misunderstanding of the very basics of Cantor's > argument (which I prefer to take in it's formalization in Z set > theory). > > There is no object in ZFC that is "Cantor's diagonal". > > Rather, for any given enumeration of a set of binary sequences, there > is a diagonal and anti-diagonal for that enumeration. > > So, your question should be, for any given enumeration of a set of > denumerable binary sequences, is the diagonal and/or the anti-diagonal > constructible?
Just that is what I expressed in short words. > > Or you might mention some particular enumeration of some particular > set of denumerable binary sequences and ask whether its diagonal and/ > or anti-diagonal is constructible? > > But the answer would depend on a *ZFC* definition of 'constructible'.
And that is not possible to obtain?
> > Cantor > > uses some x of R and shows that R is an uncountable set by > > constructing an x_0 of R.
> Also we show that there is a bijection between the real interval [0 1] > and the set of denumerable binary sequences. So [0 1] is uncountable. > > And if R is countable then, a fortiorti, [0 1] is countable.
Hence you agree that uncefinable, unconstructable and so on reals muts exist. > > Are you or are you not familiar with various certain ordinary Z theory > formalizations of Cantor's argument?
I am, but I do not want to discuss in Z here, but I am interested in its results. > > Please answer that question. I cannot address you very well on this > subject if I don't know whether you are or are not familiar with some > treatment in which Cantor's argument is formalized in Z set theory. > > > We can show, probably also in ZFC, that the constructible elements of > > R are not an uncountable set. > > That requires your definition IN THE LANGUAGE of ZFC of > 'constructible'. > > Do you know what is meant by 'the language of ZFC'? > > Please answer that question, as again, I need to know in order to > guage my discussion with you.
Yes, I know. But I do not want to use it. I only want to see the question below answered:
> > > ZFC proves a FORMULA that we READ or RENDER in ENGLISH as "there > > > exists an x such that x is uncountable". > > > This proof is probably done by constructing another element called the > > anti-diagonal? > > Actually, there is another proof in which the anti-diagonal method is > not used.
If you mean Hessenberg's proof, then let it aside for a moment. It would deserve further but different discussion. > > And again, just to be clear, there is not THE anti-diagonal, but > rather there is the anti-diagonal for each given enumeration of a set > of denumerable binary sequences.
And how many such given enumerations of a set of denumerable binary sequences do exist?
> > > So, concerning the constructible elements alone, ZFC cannot be able to > > prove their countability? Or how else could you avoid a contradiction? > > Please, you're wearing my patience. > > I have no doubt that you may show whatever contradictions between ZFC > and your own notions. That is not at issue with me. > > On the other hand, you have not shown any ZFC derivation of a formula > P & ~P nor have you shown that one exists.
I am not about to show that. I am only interested in the question: Can ZFC define "constructible reals"? Can it prove their countability?