Date: Dec 15, 2012 7:33 PM Author: Virgil Subject: Re: On the infinite binary Tree 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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