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Topic: Finitely definable reals.
Replies: 52   Last Post: Jan 18, 2013 2:37 PM

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 Virgil Posts: 8,833 Registered: 1/6/11
Re: Finitely definable reals.
Posted: Jan 15, 2013 3:06 PM

In article
WM <mueckenh@rz.fh-augsburg.de> wrote:

> On 15 Jan., 18:01, Dick <DBatche...@aol.com> wrote:
> > >On Monday, January 14,  You are right, not all finite strings are
> > >>meaningful. But all meaningful strings are a subset of all finite
> > >>strings. That is sufficient to know that all meaningful strings are
> > >>countable. Regards, WM

> >
> > All meaningful strings that define real numbers are countable in the sense
> > that they can be mapped into the integers. However, it is not true that
> > they are countable in the sense that they can be written in a list.

>
> > Suppose that there were some way to lok at a statement and determine that
> > if it was meaninful or not. Then you could write the meaningful statements
> > in a list. Statement "Diagonalize the set of numbers defined by this list"
> > is a meaninful statement that defines a real number; it should be on the
> > list!

>
> > The simplest way to resolve this paradox is to accept that Vountability (
> > mappable to the integers) and countability (writable in a list) are
> > different concepts.

>
> I think the simplest way to resolve that paradoxon is to accept that
> infinity is infinite. Of course the number defined by the diagonal can
> be appended. Then we get another list, and that may have another
> diagonal. It is simply humbug to think that the natural numbers can be
> exhausted by any procedure.

The induction procedure exhausts both the first natural and also the
successor of every natural, so what naturals does it fail to exhaust?
>
> The whole Cantor-argument fails if we recognize that every anti-
> diagonal must differ at a *finite* position from the entries of the
> list.

At each finite position, the diagonal only needs to differ from one list
entry, a different one for each of its infinitely many finite positions.

> But we can set up a list with all possible finite initial
> segments that contain all possible finite positions. Therefore it is
> not possible that the anti-diagonal differs from all entries of the
> list at a finite position.

The anti-diagonal only has to differ from one list entry at any one
position and a different one at each of its potions, and as the
anti-diagonal has more than any finite number of positions, it will
differ from more that any finite number of entries.

> And it helps nothing that the anti-doagonal
> is infinite, because either it differs at a finite position or it
> differs nowehere.

The infiniteness of the number of its positions helps because it can
differ from a different list entry at each of its infinitely many
different postitions, at last in standard mathematics, even if not in
WMytheology.
>
> In effect Cantor has proven that the countable set of finite initial
> segments (subset of the rationals) is uncountable.

Only in WMytheology is an non-finite sequnce also counted as a finite
initial segment.
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