Date: Dec 13, 2012 2:28 PM
Author: Virgil
Subject: Re: On the infinite binary Tree

In article 
<661c229a-6737-46c9-888c-d88cbb54a707@4g2000yqv.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:

> On 13 Dez., 11:49, Zuhair <zaljo...@gmail.com> wrote:
> > On Dec 13, 9:56 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> >
> >

> > > It is nonsensical because the same could be assumed for Cantor's
> > > diagonal. It would be undefinable and it would be impossible to prove
> > > that it differs from all lines of the list - in particular if
> > > undefinable reals exist and are members of the list.

> >
> > Of course the diagonal in some cases can be non-definable, that is
> > well known.

>
> Also the elements of the list could be non-definable, if non-definable
> real numbers existed.


Note that no list of defineable binary sequences (or of defineable by
digit sequence real numbers, for that matter) can be complete because
any such list defines at least one more not in that list, so it is d
sets of defineable binaries and and sets of defineable reals whuch
either cannot exist at all or cannot be countable.
>
> > That doesn't mean that we cannot prove it is different from all reals
> > in the list, on the contrary we don't need parameter free definability
> > in order to determine that the diagonal is different from the reals in
> > the original list, we can do that without it, as Cantor did.

>
> Cantor did not accept non-definable reals. If he had, he would have
> seen that his proof fails.


But any set of all defineable-by-digit-sequences-in-base-n reals either
cannot exist at all or turns out to be uncountable.
> >
> > Your error is that you think too much of non definability. It is not
> > so destructive as you think.

>
> No? Nearly every real number is undefinable. The measure of definable
> reals is 0. If most reals are non-definable, why must all reals of
> every Cantor list always be definable? If all reals of the list are
> definable, then they belong to a countable set. Then we cannot prove
> uncounatbility. Or can we prove that the set of definable reals is
> uncountable - because it is countable but there are, somewhere else,
> undefinable "reals"?


But any set of all defineable-by-digit-sequences-in-base-n reals either
cannot exist at all or turns out to be uncountable.
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