```Date: Dec 14, 2012 9:38 AM
Author: Zaljohar@gmail.com
Subject: Re: On the infinite binary Tree

On Dec 14, 5:30 pm, Zuhair <zaljo...@gmail.com> wrote:> 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, 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 ONE-ONE, 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 formulaswhich is....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.>> 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> ONE-ONE. 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
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