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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 12, 2013 4:47 PM

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

> On 11 Jan., 23:30, Virgil <vir...@ligriv.com> wrote:
>
>

> > > But my question aimed at the application of undefined reals in
> > > mathematics.

> >
> > Without them one cannot have a complete infinite Archimedean ordered
> > field such as the real number field.

>
> And with them one cannot have it either.

Perhaps not in WMytheology, but one can most other places.

> Try to discern a real that is not in the Binary Tree constructed from
> a countable set.

I do not find any reals IN binary trees.
>
>

> > > No, you misunderstand again. Cantor's opinion was (and did not change
> > > until he died) that undefined items are nonsense. And ofcourse he was
> > > absolutely right.

> >
> > But he still showed that the set of real numbers, i.e., the objects
> > forming the unique Archimedean complete totally ordered field was not a
> > countable set in the sense that no surjection from |N to that set is
> > possible.

>
> He assumed that a set of all naturals exist, which is an assumption as
> wrong as the assumption that a set of naturals can have cardinality 10
> and sum 10.

It is only "wrong" in dim dark places like WMYTHEOLOGY.
>
>

> > One may know that a real is between 0.1 and 0.2 but still not finitely
> > definable. In fact one may know a real accurate to any finite number of
> > decimals places but still have it undefineable any further.

>
> which is tantamount to *not* having any real number but only a
> rational interval.

So, according to WM, one can have an interval without having any of its
members? Typical!
>
> >
> > So if you only know its first n digits, that number is one of those
> > undefineables that WM claims do nt exist..

>
> No, there is no number known. Of course there are definable rationals
> and reals in the interval.

Then one has at least one number.
>
>

> > > There are two cases:
> > > 1) If a Cantor list is finitely defined, then you know the entry in
> > > every line and you know every digit of the diagonal.

> >
> > So that every list of finitely defined basal numerals, with base >=4,
> > is incomplete since its antidiagonal is not listed.

>
> Not a list which contains all (terminating) representations of
> rationals.

Such a list is trivially incomplete without ever worrying about
antidiagonals, since there is no base in which every rational has a
terminating represention, much less very real.
> >
> > > 2) If a Cantor list is undefined and has only, as usual, the first
> > > three lines and then an "and so on", then you do neither know the
> > > following entries nor the digits of the diagonal. Nothing is
> > > "discernible" then except the theorem that two decimals which differ
> > > at some place are not identical. But that is not a deep recognition.

> >
> > But any assertion that a list of basal numerals is COMPLETE is
> > falsified by the existence of anti-diagonals which are provably not in
> > the original list.

>
> It is already falsified by the non-existence of the sets of naturals
> mentioned above.

I see no such mention.
>
> Regards, WM

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