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

 Messages: [ Previous | Next ]
 mueckenh@rz.fh-augsburg.de Posts: 18,076 Registered: 1/29/05
Re: Finitely definable reals.
Posted: Jan 12, 2013 3:20 AM

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

> > 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.

> 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 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.

> > 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.
>
> > 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.

Regards, WM