
Re: UNCOUNTABILITY
Posted:
Dec 21, 2012 6:04 AM


On 20 Dez., 19:32, Zuhair <zaljo...@gmail.com> wrote: > On Dec 20, 4:58 pm, WM <mueck...@rz.fhaugsburg.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, not cranks as you, Greene or Knox of course. > > The bijection between all finite words and the natural numbers is NOT > parameter free definable.
It need not to be defined. I is proved (iff a complete set of natural numbers is assumed).
> 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.
This bijection is provable (iff a complete set of natural numbers is assumed).
> 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.
There is no "natural sense", but a proof. And there is no mathematician refuting that. Take for instance:
If we pursue the thought that each real number is defined by an arithmetical law, the idea of the totality of real numbers is no longer indispensable. (Bernays)
Definiert man die reellen Zahlen in einem streng formalen System, in dem nur endliche Herleitungen und festgelegte Grundzeichen zugelassen werden, so lassen sich diese reellen Zahlen gewiß abzählen, weil ja die Formeln und die Herleitungen auf Grund ihrer konstruktiven Erklärungen abzählbar sind. (Schütte)
> 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 they are not definable with parameters either.
> 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.
You cannot define something by undefined notions. The universe of discourse is for you something like the heaven is for religious people. But mathematics is not religion. > > Do you think that the bijection between the naturals and all finite > words parameter free definable?
I am not interested in the question whether it is definable, but I know that this bijection can be proved, for instance by the following list: 0 1 00 01 10 11 000 ...
> 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.
That does only prove that matheologians have a blind spot.
Proof: Cantor assumed a list that is enumerated by all natural numbers, i.e., which is complete with respect to its enumeration. Then he "proved" that there is always a real number that is not in the list. Then he proved that this real number can be inserted as another entry into the list without removing another entry. The new list is again complete and does not contain more entries than before. In effect he proved that the first list had not been complete with respect to the enumeration, i.e., he proved that his crucial assumption had been wrong. But as every logician knows, from a wrong assumption everything can be concluded.
Regards, WM

