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Topic: Finitely definable reals.
Replies: 52   Last Post: Jan 18, 2013 2:37 PM

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 Zaljohar@gmail.com Posts: 2,665 Registered: 6/29/07
Re: Finitely definable reals.
Posted: Jan 11, 2013 6:36 AM

On Jan 11, 12:49 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 11 Jan., 10:16, Zuhair <zaljo...@gmail.com> wrote:
>

> > Lets say that a real r is finitely definable iff there is a predicate
> > P that is describable by a Finitary formula that is uniquely satisfied
> > by r.

>
> Let's say a real r is finitely definable if every mathematician can
> understand the definition. The common understanding of two
> mathematicians can be proved by the identity of their decimal
> expansions of r. And via induction we can prove the common
> understanding of n mathematicians.
>
>
>

> > Of course NOT all reals are finitely definable in the above manner.
>
> Of course a real is only what satisfies this criterion.
>

> > Also it is obvious that we have only COUNTABLY many finitely definable
> > reals.

>
> So to speak, we have only countably many real reals. Why should we
>
>
>

> > Other kinds of reals can be "infinitely" definable,
>
> that means undefinable
>

> > this can be
> > achieved in a language that encounters infinitely long strings of
> > symbols,

>
> and, therefore, is not a language.
>

> > and many known first order languages are of that sort and
> > they are proven to be consistent and even supportive of a proof
> > system.

>
> But they can only be understood by very patient listeners. Up to now
> not even one single word has ever beenb understood, unless it has been
> encoded by a finite word.
>
> In any case this kind of nonsense has nothing to do with Cantor who
> held the opinion that infinite words are nonsense.
>
>
>

> > However one must understand that when we say that we have countably
> > many finitely definable reals then we are accepting the existence of a
> > bijection between the naturals and the finitely definable reals and
> > that this bijection is itself not finitely definable!

>
> But that is of no interest. It only helps to vail the strong
>

> > This is also a
> > corollary of Cantor's arguments.

>
> Cantor would have been surprised if not angry. He held the opinion
> that infinite words are nonsense.
>

> >Also the diagonal on the list of all
> > finitely definable reals IS also non finitely definable real! since it
> > is defined after the bijection between the naturals and the set of all
> > finitely definable reals, and that bijection as said above is not
> > finitely definable.

>
> So there are countable sets that are not in bijection with |N. And
> uncountable sets which are in bijection with |N, namely the set of all
> distinguishable real numbers. The undistinguishable real numbers
> cannot be distingusihed and count only as one number. This number may
> be put in bijection with zero.
>
>
>

> > Finitely definable reals are definitely very interesting kinds of
> > reals,

>
> The< are the only kind of interesting reals, because the others are
> not distinguishable, hence at most one of them can appear in counting
> the reals. In mathematics, as solutions of equationsfor instance, no
> undefinable real can ever appear.
>

> >  they are superior to those that are non finitely definable of
> > course,

>
> of course! very! In particulat because the non-finitely definable,
> i.e., in every discourse undefinable reals, cannot have any other
> mathematical properties than being undefinable.
>

> > but however that doesn't mean that the later ones do not
> > exist,

>
> Where and in what form could they exist? Does God know them? Can He
> distinguish them?
>

> > nor does it mean that the later ones cannot be spoken about, we
> > can still speak of those kinds of reals

>
> You can speak about the set or a typical element, but you cannot prove
> mathematically, that there are more than one. Remember: They do not
> occur in mathematics.
>

> > by using formulas that do not
> > uniquely hold of one of them, and still those sentences can illustrate
> > interesting pieces of mathematics that might possibly find some
> > application one day.

>
> But unfortunately that day will not appear in the first eternity.
>

> > However it is expected of course that finitely
> > definable reals would be of more importance no doubt and therefore
> > they would have the leading stance among reals.

>
> Please answer one question: What shall undefinable reals be good for?

Explaining continuity of space? Possibly?

> They cannot spring off Cantor's argument.

They do of course, they are a consequence of his arguments.

Cantor proved that the
> *definable* reals (those which are definitely different from all reals
> of his list) cannot be put in bijection with |N.

You mean * discernible* reals, there is a difference between
discernible reals and finitely definable reals, two reals might be
discernible (i.e. differ at some finite position of their decimal
expansions) and yet each one of them might be non finitely definable!
YES Cantor proved that his Diagonal real is * discernible* from all
the other members of the list, AS FAR AS THAT LIST IS COUTNABLE, but
that doesn't make out of it *finitely definable*; for it to be
finitely definable it must UNIQUELY satisfy some finite predicate and
proving it discernible doesn't by itself make out of it finitely
definable. Cantor's arguments tells us that we do have MORE
discernible reals than finitely definable ones. We do have UNCOUNTABLY
many discernible reals but we have only COUNTABLY many finitely
definable reals.

But we know that they
> are countable. Undefinable reals are not elements of mathematics and
> of Cantor-lists. They cannot help to make the defined diagonals belong
> to an uncountable set.
>

No some of Non finitely definable reals ARE members of Cantor-lists.
Actually for some lists the diagonala is provabley (by Cantor's
arguments) non finitely definable! But of course all elements on
Cantor's list and the diagonal (or antidiagonal) all are definitely
discernible (i.e. differ from each OTHER real at some finite position
of their decimal expansions).

You are confusing * discern-ability* with *finite definability*

> Regards, WM