On Dec 20, 4:58 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > On 19 Dez., 22:11, Zuhair <zaljo...@gmail.com> wrote: > > > Thus the set of all parameter free definable reals is UNCOUNTABLE! > > QED > > And it is countable by the simple proof that there is a bijection > between all finite words an the natural numbers. Hence the result of > your argument is not that the other one is wrong but that set theory > is inconsistent, namely the notion of countability that presupposes > finished infinity is contradictory, as everybody with a sober mind > would see immediately, nit cranks as you, Greene or Knox of course.
The bijection between all finite words and the natural numbers is NOT parameter free definable. So if one insist that EVERY set must be parameter free definable, then this mean that there is no bijection between the set of all finite words and the naturals. HOWEVER I already said that this result only stems if one desires that ALL sets must be parameter free definable, which is as I said a high price to pay, because there is a natural sense of the existence of a bijection between all naturals and all finite words. NOW if we follow that natural sense then this mean that we must give up the concept that all sets are parameter free definable, because any bijection between the naturals and the set of all finite words is itself NOT parameter free definable. And this opens the door wide for accepting sets that are not parameter free definable. And of course the number of those sets is determined by the number of assignments given to parameters in the defining formulas, which is something that range over the whole universe of discourse, so it is not limited by the countability of those formulas.
Do you think that the bijection between the naturals and all finite words parameter free definable?
> > > So having parameters in the defining formulas provides the grounds for > > POSSIBILITY of having uncountably many sets definable after them. But > > what PROVES the existence of uncountabily many parameter definable > > reals is of course the diagonal argument of Cantor. > > > Anyway we turn matters we face uncountability! > > > The alternative explanation that uncountability only rise from local > > defect in the expressive language of a theory and that all sets are > > countable in the real world, though possible yet looks far from being > > the case. The only cause for believing in such a direction is a bias > > towards countability, > > You are dreaming of parameters that help to define uncountably many > elements. But that is and remains a dream, unless you assume the > existence of uncountably many undefined parameters. That, however, is > matheology, not mathematics. Since parameters do not help anything in > defining uncountably many elements, set theory is contradicted. > Parameters helps in opening the *POSSIBILITY* of having uncountably many elements. IT doesn't not prove it. What proves uncountability of the universe of discourse is Cantor's diagonal argument.