Virgil
Posts:
8,833
Registered:
1/6/11


Re: On the infinite binary Tree
Posted:
Dec 15, 2012 7:33 PM


In article <5ecbfa16e85e4d97b4ef6e80a8e036b7@b8g2000yqh.googlegroups.com>, WM <mueckenh@rz.fhaugsburg.de> wrote:
> On 14 Dez., 15:30, Zuhair <zaljo...@gmail.com> wrote: > > On Dec 14, 12:32 am, WM <mueck...@rz.fhaugsburg.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 ONEONE, it can be MANYONE. > > > > 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 MANYONE 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 

