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

 Messages: [ Previous | Next ]
 Virgil Posts: 8,833 Registered: 1/6/11
Re: Finitely definable reals.
Posted: Jan 11, 2013 4:47 PM

In article
WM <mueckenh@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.

That would make reals language-dependent!

> The common understanding of two
> mathematicians

Not if WM gets to decide who qualifies as a mathematician.

> can be proved by the identity of their decimal
> expansions of r.

It may well be that a real number does not have any complete decimal
expansion, but has other definition on which people can agree

> And via induction we can prove the common
> understanding of n mathematicians.

Nonsense! For one thing, there are a lot of mathematicians who would
deny WM membership in that group.
> >
> > Of course NOT all reals are finitely definable in the above manner.

>
> Of course a real is only what satisfies this criterion.

Only in Wolkenmuekenheim. Elsewhere, one goes by more reliable
definitions.
>
> > 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
> bother about unreals?

WM may see only countably many, but his mathematical vision is
well-known to be deficient.
> >
> > Other kinds of reals can be "infinitely" definable,

>
> that means undefinable

But even if undefined, not necessarily non-existent.
>
> > this can be
> > achieved in a language that encounters infinitely long strings of
> > symbols,

>
> and, therefore, is not a language.

At least not one that WM could possibly understand, but there is a great
deal of mathematics that WM has already shown that he does not and
probably can not, understand.
>
> > 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.

Which WM is not.

>
> In any case this kind of nonsense has nothing to do with Cantor who
> held the opinion that infinite words are nonsense.

But Cantor also managed to prove that there are more than countably many
infinite binary strings possible and that the standard definition of the
standard complete ordered real number field requires the existence of
more than countably many real numbers.
> >
> > 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.

A lot of thing are of interest to most mathematicians which are not of
interest to WM, and vice versa.

> It only helps to vail the strong
> contradictions of matheology.

That is spelled "WMytheology".
>
> > 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.

Nowhere nearly as nonsensical as he would have found WM's claims.
>
> >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.

Of course! All finite sets are countable but none have bijections with
|N.

> And
> uncountable sets which are in bijection with |N, namely the set of all
> distinguishable real numbers.

Being in bijection with |N is a special case of being countable.

At least in standard mathematics, where the definition of a set being
countable is that there is a surjection from N to that set.
So perhaps in WMytheology bijections need not be surjections.

> The undistinguishable real numbers
> cannot be distingusihed and count only as one number.

Only in Wolkenmuekenheim.

> This number may
> be put in bijection with zero.

Only in Wolkenmuekenheim.
> >
> > 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.

But without those undefinable reals one cannot have a complete ordered
Archimedean field, as requirements of such a field cannot be met by any
countable set of reals.
>
> > 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.

While they many not occur in WMytheology, they do occur in any complete
ordered Archimedean field like the Real NUmber Field.
>
> > 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?

Filling in the gaps between the defineable ones so as to have a complete
ordered Archimedean field in which every set of reals which is bounded
above has a least upper bound and every Cauchy sequence has a real
number limit.

> They cannot spring off Cantor's argument. Cantor proved that the
> *definable* reals (those which are definitely different from all reals
> of his list) cannot be put in bijection with |N. But we know that they
> are countable.

Since the definition of a set, S, being countable requires the
existence of a surjection from |N to S, WM is claimimg the existence of
a set D of definable reals such that there is a surjection from |N to D
(because that is the definition of countability) but no bijection from
|N to D.

A little thought shows that the only sets D such that |N surjects to D
but does biject with D, are finite sets.

So that in Wolkenmuekenheim the set of all definable reals is a finite
set.
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