Date: Dec 13, 2012 4:19 PM
Subject: Re: On the infinite binary Tree

On 13 Dez., 20:47, Zuhair <> wrote:

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

> No Cantor's proof survives non parameter free definability. We don't
> need every real to be definable by a parameter free formula in order
> for Cantor's proof to go through.

Wrong. Undefinable reals are undefinable. Completely undefined. But
that is of little interest. Our concern is that the Binary Tree
contradicts Cantor's proof.

> That's your simple mistake, you
> think Cantor's proof requires that all reals must be parameter free
> definable, but this is not the case. Cantor's proof works in a
> flawless manner even if MOST of the reals are non parameter free
> definable.

At least if we want to know the diagonal, we need every line to be
excplicitly defined.
And in fact most of the reals are undefinable.

> Actually Cantor's proof mounts to the conclusion that MOST
> reals are non parameter free definable reals, of course he saw that,
> this is obvious really.

Obvious is only that you have no idea of that matter. Cantor was not
horrified by the idea that all definable reals belong to a countable
set because he thought that this idea is wrong.

Fact is: If all list entries are definable and the diagonal is
definable too, then Cantor's proof proves the uncountability of
definable reals. If not all list entries are definable, then there is
no diagonal at all.

Regards, WM