Date: Jan 12, 2013 4:47 PM Author: Virgil Subject: Re: Finitely definable reals. In article

<42e94422-7b70-4058-88d7-0035491b19c1@f4g2000yqh.googlegroups.com>,

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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