Virgil
Posts:
9,012
Registered:
1/6/11


Re: Distinguishability of paths of the Infinite Binary tree???
Posted:
Dec 29, 2012 2:21 AM


In article <577c1bed657f4aa2a66c9d8f225fdcff@gu9g2000vbb.googlegroups.com>, Zuhair <zaljohar@gmail.com> wrote:
> On Dec 29, 4:06 am, Virgil <vir...@ligriv.com> wrote: > > In article > > <51eb8729b50a41368af62c527c93b...@x20g2000vbf.googlegroups.com>, > > > > Zuhair <zaljo...@gmail.com> wrote: > > > By the way you said it is not obvious to you what I meant by parameter > > > free definable > > > > An example of a mathematical definition ( or even a non mathematical > > one) which you regard as being "parameter free" and of one which you > > regard as not being "parameter free" might clear the air. > > > > I am not totally clear in my owm mind what you mean by the phrase. > > > > Do you, for instance, regard the standard definition of countability of > > a set (set S is countably if and only if there exists a surjection from > > N to S) as being parameer free or not, and why? > >  > > This is a well known subject. There is no problem if some people don't > know about it. The problem is if some people professing big claims > like refuting Cantor or saying that THOUSANDS of mathematician for a > CENTURY long time are acting fools, and then it turns that those > people themselves don't know basic definitions? really strange! > > Now we come to your example: > > Is countability of a set parameter free definable or not? > > The answer is YES. > > Why? > > Because there is a parameter free formula "phi(S)" such that > > For all S. Countable(S) iff phi(S) > > And what is meant by phi(S) being parameter free is that phi(S) is a > formula in which only the symbol S occurs free, i.e. all other > variable symbols in phi(S) are quantified within the formula, and of > course S is free. > > Now lets explicitly examine this > > take phi(S) to be the following formula > > Exist f. Exist N. (for all y. y in N iff y is a finite ordinal) & f: S > > N & f is injective. > > The open expansion of this formula show that only the variable symbol > S occurs free. > QED > > So for example in a theory like NF where we do have the set of all > countable sets. So this set is parameter free definable set, because > membership of this set follows satisfaction of a parameter free > formula. > > Now let me give you an example of an object that cannot be definable > in a parameter free manner. > Lets take some theory that provides sufficient material to define > 'definable real' as: > > x is a definable real <> iff x is a real & Exist phi. for all y. y in > x <> phi(y) > > where phi(y) is a parameter free formula (i.e. only y occurs free in > phi(y)). > > Now one can prove using Cantor's argument that ANY bijection F between > the set R* of ALL definable reals and the set N of all naturals IS non > parameter free definable > > The reason is that if we suppose the contrary i.e. the existence of > such a bijection that is parameter free definable, then the diagonal > defined after it would be a parameter free definable real that is not > in the set of ALL definable reals, which is a clear contradiction. > > So although there is a bijection between R* and N, yet it is provable > that any such bijection is non parameter free definable! > > In other words there is no parameter free formula phi(y) such that the > above bijection have all its membership determined after satisfaction > of phi(y). > > Hope that is helpful and clear > > > Zuhair
Largely both. Thanks. 

