Virgil
Posts:
8,833
Registered:
1/6/11


Re: On the infinite binary Tree
Posted:
Dec 13, 2012 2:28 PM


In article <661c229a673746c9888cd88cbb54a707@4g2000yqv.googlegroups.com>, WM <mueckenh@rz.fhaugsburg.de> wrote:
> On 13 Dez., 11:49, Zuhair <zaljo...@gmail.com> wrote: > > On Dec 13, 9:56 am, WM <mueck...@rz.fhaugsburg.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 nondefinable, that is > > well known. > > Also the elements of the list could be nondefinable, if nondefinable > 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 nondefinable reals. If he had, he would have > seen that his proof fails.
But any set of all defineablebydigitsequencesinbasen 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 nondefinable, 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 defineablebydigitsequencesinbasen reals either cannot exist at all or turns out to be uncountable. 

