```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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