Date: Dec 13, 2012 6:49 PM
Author: Virgil
Subject: Re: On the infinite binary Tree
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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