Virgil
Posts:
4,483
Registered:
1/6/11
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Re: Distinguishability of paths of the Infinite Binary tree???
Posted:
Dec 29, 2012 2:21 AM
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In article
<577c1bed-657f-4aa2-a66c-9d8f225fdcff@gu9g2000vbb.googlegroups.com>,
Zuhair <zaljohar@gmail.com> wrote:
> On Dec 29, 4:06 am, Virgil <vir...@ligriv.com> wrote:
> > In article
> > <51eb8729-b50a-4136-8af6-2c527c93b...@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.
--
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