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Topic: Why can no one in sci.math understand my simple point?
Replies: 660   Last Post: Dec 7, 2017 8:06 AM

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 mueckenh@rz.fh-augsburg.de Posts: 18,076 Registered: 1/29/05
Re: Why can no one in sci.math understand my simple point?
Posted: Jun 18, 2010 1:03 PM

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.
>
> 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

> > > 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?

Regards, WM

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6/17/10 mueckenh@rz.fh-augsburg.de
6/17/10 Virgil
6/17/10 Jack Markan
6/17/10 mueckenh@rz.fh-augsburg.de
6/17/10 Virgil
6/17/10 Jack Markan
6/17/10 Jack Markan
6/17/10 lwalke3@lausd.net
2/17/13
12/7/17 4musatov@gmail.com
6/18/10 mueckenh@rz.fh-augsburg.de
6/18/10 Virgil
6/17/10 Aatu Koskensilta
6/17/10 mueckenh@rz.fh-augsburg.de
6/17/10 lwalke3@lausd.net
6/17/10 Virgil
6/17/10 Sylvia Else
6/17/10 mueckenh@rz.fh-augsburg.de
6/17/10 Sylvia Else
6/17/10 lwalke3@lausd.net
6/17/10 |-|ercules
6/17/10 Sylvia Else
6/18/10 |-|ercules
6/18/10 Sylvia Else
6/18/10 |-|ercules
6/18/10 Sylvia Else
6/18/10 |-|ercules
6/18/10 Sylvia Else
6/18/10 mueckenh@rz.fh-augsburg.de
6/18/10 Sylvia Else
6/18/10 |-|ercules
6/18/10 Sylvia Else
6/18/10 |-|ercules
6/19/10 Sylvia Else
6/19/10 |-|ercules
6/19/10 Sylvia Else
6/19/10 |-|ercules
6/19/10 Sylvia Else
6/19/10 |-|ercules
6/20/10 Sylvia Else
6/20/10 |-|ercules
6/19/10 mueckenh@rz.fh-augsburg.de
6/19/10 Virgil
6/18/10 Virgil
6/18/10 mueckenh@rz.fh-augsburg.de
6/18/10 Virgil
6/18/10 |-|ercules
6/18/10 Sylvia Else
6/19/10 |-|ercules
6/19/10 Sylvia Else
6/19/10 |-|ercules
6/19/10 Sylvia Else
6/19/10 |-|ercules
6/20/10 Sylvia Else
6/20/10 |-|ercules
6/20/10 Sylvia Else
6/20/10 |-|ercules
6/20/10 lwalke3@lausd.net
6/20/10 |-|ercules
6/20/10 |-|ercules
6/19/10 mueckenh@rz.fh-augsburg.de
6/19/10 Virgil
6/19/10 mueckenh@rz.fh-augsburg.de
6/19/10 Virgil
6/18/10 mueckenh@rz.fh-augsburg.de
6/18/10 Virgil
6/19/10 mueckenh@rz.fh-augsburg.de
6/19/10 Virgil
6/18/10 ostap_bender_1900@hotmail.com
6/21/10 Sylvia Else
6/22/10 Sylvia Else
6/27/10 Frederick Williams