
Re: On the infinite binary Tree
Posted:
Dec 14, 2012 9:38 AM


On Dec 14, 5:30 pm, Zuhair <zaljo...@gmail.com> wrote: > On Dec 14, 12:32 am, WM <mueck...@rz.fhaugsburg.de> wrote: > > > > > > > > > > > On 13 Dez., 21:02, Zuhair <zaljo...@gmail.com> wrote: > > > > On Dec 13, 3:56 pm, WM <mueck...@rz.fhaugsburg.de> wrote: > > > > > No? Nearly every real number is undefinable. The measure of definable > > > > reals is 0. If most reals are nondefinable, why must all reals of > > > > every Cantor list always be definable? If all reals of the list are > > > > definable, then they belong to a countable set. Then we cannot prove > > > > uncounatbility. Or can we prove that the set of definable reals is > > > > uncountable  because it is countable but there are, somewhere else, > > > > undefinable "reals"? > > > > > Regards, WM > > > > Cantor's list do contain non definable reals. > > > Which one in what line? What is the corresponding digit of the > > diagonal? > > > > Actually some diagonals > > > of Cantor's are non definable. > > > Lists containing undefinable entries do not supply diagonals at all. > > > > The bijective function between all > > > definable reals and the set N of all naturals is NON definable set! > > > That is completely without interest. > > > > Note: definable is short for "definable by parameter free finite > > > formula" > > > No. Definable means "definable by a finite word". Everything else is > > "undefinable". > > > Regards, WM > > 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. > > While to you it seems you mean a real that is definable after some > finite formula. > > These are two different concepts, and we do need to look into those > carefully. > > Generally speaking a set X is called Definable iff there is a finitary > formula phi such that > for all y. y in X <> phi > > Now a set X is called "parameter free definable set" iff there is a > finitary formula phi that is parameter free such that for all y. y in > X <> phi > > However customarily speaking when we say definable mostly it refers to > parameter free definability, anyhow. > > So for the sake of discussion here I will give the term "definable" > for any finitary formula, and I'll use > "parameter free definable" if the defining formula is a finite formula > that is parameter free, as depicted above. > > Now there is an important question: Is all reals parameter free > definable? > If the answer is yes then it is clear that we will have COUNTABLY many > reals, since we have countably many parameter free finite formulas, > that's obvious. > > One needs to be careful here, if a real is not parameter free > definable that doesn't mean it is not definable! since there can be a > finitary formula that is not parameter free after which it can be > defined. A famous example is the diagonal after a bijective function > from the set of all naturals to the set of all parameter free > definable reals. This diagonal is DEFINABLE but not in a parameter > free manner. > > There is something very important that people need to realize about > the difference between definability and parameter free definability. > which is the following: > > If one say that all reals are parameter free definable then this leads > to saying that all reals are countable! > > If one say that all reals are "definable" then this does NOT lead to > saying that all reals are countable!!! > > WHY? > > One would wonder saying what is the difference, in any case we have > countably many formulas whether they are parameter free or not? so for > the first glance it seems that what make all parameter free definable > reals countable is the same thing that causes one to believe that all > definable reals are countable, and obviously that thing is the > countability of all finitary formulas. > > But the reality of the matter is that they differ greatly and the > above reasoning is not correct. The presence of a parameter in the > defining formula (which is of course finite as stipulated above) will > make a big difference from its absence. > > Lets take the following example to show that difference: > > For all y. y in X <> y in A & pi(y) > > so the defining formula of X here is of course the formula "y in A & > pi(y)" where pi(y) is some > formula that is finitary parameter free formula. > > Now the formula "y in A & pi(y)" is a finitary formula but it is not > parameter free, it contains a parameter and that parameter is A. > > Lets go more concretely and lets stipulate pi(y) to be the formula "y > is an even number" > > Now how many sets X do I have that can be defined in the above manner? > The answer depends clearly on how many sets the symbol A can range > over. > > Suppose for the sake of discussion that we have uncountably many sets > that A can range over. > Now for each substitution of A by a set we have a set X that > correspond to that substitution, so it is possible to have uncountably > many sets X defined from uncountably many substitutions of the > variable A in the SINGLE formula "y in A & pi(y)". > > So we can define MANY sets after the single formula which is " y in A > & pi(y) " > > So here the relationship between the number of sets defined after a > formula having a parameter is not ONEONE, no it is actually MANY  > ONE relationship. > > 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,
Oops, I meant ...even if we have countably many finitary formulas which is....
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. > > However the situation differs for "parameter free definable" reals. > Here matters are completely different. Lets come back again and > analyse matters. > > For all y. y in X <> phi(y) > > phi(y) is parameter free (i.e. have no free variables other than y). > > Now because of Extensionality, we know that there is only ONE set X > than can be defined after EACH formula phi(y). So the relationship > between the defined sets and the parameter free defining formulas is > ONEONE. 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. > > 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! > > Zuhair

