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Re: On the infinite binary Tree
Posted:
Dec 13, 2012 7:56 AM
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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.
> 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.
>
> 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"?
Regards, WM
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