Virgil
Posts:
4,480
Registered:
1/6/11
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Re: On the infinite binary Tree
Posted:
Dec 15, 2012 7:33 PM
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In article
<5ecbfa16-e85e-4d97-b4ef-6e80a8e036b7@b8g2000yqh.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:
> On 14 Dez., 15:30, Zuhair <zaljo...@gmail.com> wrote:
> > On Dec 14, 12:32 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> >
> >
>
> > > > Cantor's list do contain non definable reals.
> >
> > > Which one in what line? What is the corresponding digit of the
> > > diagonal?
>
> No answer.
> >
> > > No. Definable means "definable by a finite word". Everything else is
> > > "undefinable".
>
>
> > Hmmm... then we are speaking about different concepts.
> >
> > For me when I say Definable real, it means real that is definable
> > after some FINITE formula that is PARAMETER FREE.
>
> That is the same. A formula is a finite word.
> >
> > While to you it seems you mean a real that is definable after some
> > finite formula.
>
> That is a finite word.
> >
> > These are two different concepts, and we do need to look into those
> > carefully.
>
> What do you think to gain by parameters?
> >
>
> > That's why even if we have countably many parameter free finitary
> > formula which is of course the case as we all know, still this doesn't
> > mean that the number of all sets definable after those formulas is
> > also countable, why because for finitary formulas that contain
> > parameters the relationship between the sets defined after those
> > formulas and those formulas is not ONE-ONE, it can be MANY-ONE.
> >
> > So we of course can have uncountably many definable reals in this
> > sense.
>
> As long as you want to define the reals, you cannot use them. Then you
> have only countaby many parameters and your MANY-ONE defines at most
> aleph_0 * aleph_0 = aleph_0 numbers.
>
> >
> > However the situation differs for "parameter free definable" reals.
>
> No it is exactly the same, namely aleph_0 reals are definable with and
> without parameters.
>
> > Here matters are completely different. Lets come back again and
> > analyse matters.
> >
> > And since we have only countably many finitary formulas and
> > parameter free formulas are all finitary by definition (see above),
> > then we will definitely have countably many parameter free definable
> > reals.
> >
> > This is a subtle difference that a lot of people usually overlook.
>
> Nonsense. How can you write so much rubbish? Don't you know that one
> cannot use that what has to be defined? And if you don't use
> uncountably many parameters, then you cannot define uncountably many
> real numbers.
> >
> > Cantor is not afraid from ALL reals being definable. But definitely
> > Cantor knew that all reals cannot be parameter free definable in a
> > finitary manner. Since the later would clearly violate his diagonal,
> > but the former does not.
> >
> > Hope that helps!
>
> It helps to see that you are not the least bit informed about Cantor
> and about set theory.
Someone blind trying, to lead someone he considers blinder into blind
allays is an amusing spectacle.
>
> Regards, WM
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