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.