Virgil
Posts:
6,972
Registered:
1/6/11


Re: Finitely definable reals.
Posted:
Jan 15, 2013 3:06 PM


In article <95ff2f7b2ea4437d85fad75391bb1656@d4g2000vbw.googlegroups.com>, WM <mueckenh@rz.fhaugsburg.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 Cantorargument 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 antidiagonal differs from all entries of the > list at a finite position.
The antidiagonal only has to differ from one list entry at any one position and a different one at each of its potions, and as the antidiagonal 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 antidoagonal > 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 nonfinite sequnce also counted as a finite initial segment. 

