```Date: Dec 14, 2012 9:30 AM
Author: Zaljohar@gmail.com
Subject: Re: On the infinite binary Tree

On Dec 14, 12:32 am, WM <mueck...@rz.fh-augsburg.de> wrote:> On 13 Dez., 21:02, Zuhair <zaljo...@gmail.com> wrote:>> > On Dec 13, 3:56 pm, WM <mueck...@rz.fh-augsburg.de> wrote:>> > > No? Nearly every real number is undefinable. The measure of definable> > > reals is 0. If most reals are non-definable, 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, WMHmmm... then we are speaking about different concepts.For me when I say Definable real, it means real that is definableafter some FINITE formula that is PARAMETER FREE.While to you it seems you mean a real that is definable after somefinite formula.These are two different concepts, and we do need to look into thosecarefully.Generally speaking a set X is called Definable iff there is a finitaryformula phi such thatfor all y. y in X <-> phiNow a set X is called "parameter free definable set" iff there is afinitary formula phi that is parameter free such that for all y. y inX <-> phiHowever customarily speaking when we say definable mostly it refers toparameter 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 formulathat is parameter free, as depicted above.Now there is an important question: Is all reals parameter freedefinable?If the answer is yes then it is clear that we will have COUNTABLY manyreals, since we have countably many parameter free finite formulas,that's obvious.One needs to be careful here, if a real is not parameter freedefinable that doesn't mean it is not definable! since there can be afinitary formula that is not parameter free after which it can bedefined. A famous example is the diagonal after a bijective functionfrom the set of all naturals to the set of all parameter freedefinable reals. This diagonal is DEFINABLE but not in a parameterfree manner.There is something very important that people need to realize aboutthe difference between definability and parameter free definability.which is the following:If one say that all reals are parameter free definable then this leadsto saying that all reals are countable!If one say that all reals are "definable" then this does NOT lead tosaying that all reals are countable!!!WHY?One would wonder saying what is the difference, in any case we havecountably many formulas whether they are parameter free or not? so forthe first glance it seems that what make all parameter free definablereals countable is the same thing that causes one to believe that alldefinable reals are countable, and obviously that thing is thecountability of all finitary formulas.But the reality of the matter is that they differ greatly and theabove reasoning is not correct. The presence of a parameter in thedefining formula (which is of course finite as stipulated above) willmake 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 someformula that is finitary parameter free formula.Now the formula "y in A & pi(y)" is a finitary formula but it is notparameter free, it contains a parameter and that parameter is A.Lets go more concretely and lets stipulate pi(y) to be the formula "yis 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 rangeover.Suppose for the sake of discussion that we have uncountably many setsthat A can range over.Now for each substitution of A by a set we have a set X thatcorrespond to that substitution, so it is possible to have uncountablymany sets X defined from uncountably many substitutions of thevariable 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 aformula having a parameter is not ONE-ONE, no it is actually MANY -ONE relationship.That's why even if we have countably many parameter free finitaryformula which is of course the case as we all know, still this doesn'tmean that the number of all sets definable after those formulas isalso countable, why because for finitary formulas that containparameters the relationship between the sets defined after thoseformulas and those formulas is not ONE-ONE, it can be MANY-ONE.So we of course can have uncountably many definable reals in thissense.However the situation differs for "parameter free definable" reals.Here matters are completely different. Lets come back again andanalyse 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 Xthan can be defined after EACH formula phi(y). So the relationshipbetween the defined sets and the parameter free defining formulas isONE-ONE. And since we have only countably many finitary formulas andparameter free formulas are all finitary by definition (see above),then we will definitely have countably many parameter free definablereals.This is a subtle difference that a lot of people usually overlook.Cantor is not afraid from ALL reals being definable. But definitelyCantor knew that all reals cannot be parameter free definable in afinitary manner. Since the later would clearly violate his diagonal,but the former does not.Hope that helps!Zuhair
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