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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 8:51 PM

In article
WM <mueckenh@rz.fh-augsburg.de> wrote:

> On 11 Jan., 15:44, Zuhair <zaljo...@gmail.com> wrote:
> > On Jan 11, 3:52 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> >

>
> > > > > They cannot spring off Cantor's argument.
> >
> > > > They do of course, they are a consequence of his arguments.
> >
> > > No, you misunderstand again. Cantor's opinion was (and did not change
> > > until he died) that undefined items are nonsense. And of course he was
> > > absolutely right.

>
> No answer. You are babbling "of course", of course, but you are unable
> to support your babblings, of course, because you don't know Cantor's
> writings. You exlude yourself from scholarly discussion.

But not from mathematica discussions.

Scholarship without valid mathematical arguments, however otherwise
erudite, such as WM keeps presenting, is of no mathematical interest.
>
> > It is easy to prove that ALL reals are discernible.
>
> It is easy to prove that two given reals are discernible. But
> undefinable reals cannot be given.

Where is it written that they have to be in order to be "discernible"?
> >
> > Now the question is: Is all reals finitely definable?
> > or Equivalently: is it the case that for each real r if r is
> > discernible then r must be finitely definable.
> > or Equivalently: Can there exist a real r that is discernible but not
> > finitely definable.

>
> Can it exist in mathematics, i.e., in the discourse, i.e., can we talk
> about it? If you believe in some remote existence outside of
> mathematics, you may do so, but I am only interested in mathematical
> discourse.

You are only interested in restricting mathematical discourse to the
> >
> > YOUR answer to that question is in the negative, i.e. you think that
> > there cannot exist a real that is discernible and not finitely
> > definable. That's your stance.
> >
> > While Cantor's arguments PROVES the existence of at least one real
> > that is discernible but not finitely definable.

>
> No, it does not. If the list is undefined, then there is no
> discernability. If the list is defined, then the diagonal is defined
> too.

Since such lists must exist even if the set of reals is uncountable, one
can speak of the diagonal of such a list without it being a discernable
number.
> >
>
> > Take the list of ALL finitely definable reals.
>
> How would you do that? These reals cannot be "listed", i.e., written
> line by line, and the list cannot be defined.

If a set cannot be listed, i.e., if no surjection from |N to the set can
be shown to exist, then it is, by definition, uncountable.

> Therefore in mathematics this list does not exist.

Therefore in Wolkenmuekenheim this list may not exist, but the
constraints of WMytheology do not hold outside it..

> all sets. But that does not make it appear in mathematics.)

In many set theories no such set occurs.
>
> > You hold that this list
> > is countable, i.e. there is a function f such that f is bijective from
> > N to the set of all finitely definable reals.

>
> No. I *can prove* that there cannot be more than countably many
> finitely defined reals.

Not to the satisfaction of those not constrained to accept WMytheology.
> >
> > Ok it is easy to prove that there can be infinitely many such
> > bijective functions.

>
> You see what a rubbish your formal "proofs" are. If a list cannot be
> defined, then there is no function.
>

> > Actually the standard is that MOST of reals are
> > not finitely definable! And yet ALL of them are discernible! i.e.

>
> "Actually accepted" is not tantamount to "generally true".
> spread such nonsense are obviously predominantly filled with mud.

Those who accept the standard definition of the real numbers as forming
a complete ordered Archimedean field are all, at least according to WM,
filled with mud.
>
> > each
> > one of them do differ from all other reals at some finite position of
> > its decimal expansion.

>
> You cannot understand that this is just a definition of uncountably
> many rationals?

On the contrary, there are only countably many rationals and there are a
lot more non-rationals,
>
> If you were right, Cantor's argument would still remain true if you
> cut every entry of the Cantor list after its n-th digit.

Which amounts to replacing all those "cut" digits with 0's

Then you see
> that the anti-diagonal differs from every entry q_n at a finite place
> q_nn. But if the list contains every rational q_n (which is possible),

That list cannot even contain 1/3 or 2/3, or any other rational whose
denominator is not of the form 2^m*5^n. for m and n in {0,1,2,3,...}.

So WM is wrong again!

> then this differing is impossible because, as I mentioned already, the
> anti-diagonal has at least one double up to every finite n. - And to
> know more than all finite digits of the anti-diagonal is not possible.

> Further it has no chance to differ at digits that have no finite
> index.

I was not aware that there were any numbers whose decimal digits did not
all have finite indices.
>
> > Anyhow I don't think that further discussion with you would be
> > fruitful.

>
> In particular since you would be forced again and again to answer this
> question, at least towards yourself. But you cannot as long as you
> believe in matheology.

Given a choice between what WM calls matheology and what the rest of us
perceive as WM's WMytheology, most of us would clearly prefer the
former. Though we perceive ours as being standard mathematics and WM's
as usually not being mathematics at all.
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