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Re: Finitely definable reals.
Posted:
Jan 15, 2013 12:51 PM
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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 whole Cantor-argument fails if we recognize that every anti-
diagonal must differ at a *finite* position from the entries of the
list. 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. And it helps nothing that the anti-doagonal
is infinite, because either it differs at a finite position or it
differs nowehere.
In effect Cantor has proven that the countable set of finite initial
segments (subset of the rationals) is uncountable.
Regards, WM
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