Virgil
Posts:
4,482
Registered:
1/6/11
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Re: On the infinite binary Tree
Posted:
Dec 13, 2012 6:49 PM
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In article
<d9d2da33-b039-4af6-bfff-6dfc8efe8857@r3g2000vbn.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> 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.
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