In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> On 13 Dez., 20:47, Zuhair <zaljo...@gmail.com> 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.
WM's allegedly complete infinite binary trees are never incomplete. Every subset of N should correspond to a different path, the one branching left at just those levels whose numbers are in the set, and right at all other levels. and N has more subsets than WM's tree has paths. > > > > 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.
Not true. The argument is perfectly valid so long as all its entries are defineable, but does not need any of them explicitely defined.
> And in fact most of the reals are undefinable.
Enough are defineable! > > > 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.
While that is true enough foe WM, it is not so in general. --