Virgil
Posts:
8,833
Registered:
1/6/11


Re: The Distinguishability argument of the Reals.
Posted:
Jan 4, 2013 4:38 PM


In article <2466aecd53094468b90d3da7727466f7@f19g2000vbv.googlegroups.com>, WM <mueckenh@rz.fhaugsburg.de> wrote:
> On 4 Jan., 13:36, gus gassmann <g...@nospam.com> wrote: > > On 03/01/2013 5:53 PM, Virgil wrote: > > > > > > > > > > > > > In article > > > <a60601d524a24501a28b84a7b1e53...@ci3g2000vbb.googlegroups.com>, > > > WM <mueck...@rz.fhaugsburg.de> wrote: > > > > >> On 3 Jan., 14:52, gus gassmann <g...@nospam.com> wrote: > > > > >>> Exactly. This is precisely what I wrote. IF you have TWO *DIFFERENT* > > >>> reals r1 and r2, then you can establish this fact in finite time. > > >>> However, if you are given two different descriptions of the *SAME* > > >>> real, > > >>> you will have problems. How do you find out that NOT exist n... in > > >>> finite time? > > > > >> Does that in any respect increase the number of real numbers? And if > > >> not, why do you mention it here? > > > > > It shows that WM considerably oversimplifies the issue of > > > distinguishing between different reals, or even different names for the > > > same reals. > > > > >>> Moreover, being able to distinguish two reals at a time has nothing at > > >>> all to do with the question of how many there are, or how to > > >>> distinguish > > >>> more than two. Your (2) uses a _different_ concept of > > >>> distinguishability. > > > > >> Being able to distinguish a real from all other reals is crucial for > > >> Cantor's argument. "Suppose you have a list of all real numbers ..." > > >> How could you falsify this statement if not by creating a real number > > >> that differs observably and provably from all entries of this list? > > > > > Actually, all that is needed in the diagonal argument is the ability > > > distinguish one real from another real, one pair of reals at a time. > > > > Exactly. The only reals that matter to Cantor's argument are the > > *countably* many that are assumed to have been written down. There is no > > need (nor indeed an effective way) to distinguish the constructed > > diagonal from *all* the potential numbers that could have been > > constructed that are not on the list, either. > > Therefore they cannot interfere with Cantor's argument and cannot > result from his procedure.
According to standard mathematics, a set is "countable" if and only if there is surjection from N to that set. And until WM can produce such a surjection, none of his claims that the reals are countable count. > > > Any *one* number not on > > the list shows that the list is incomplete and thus establishes the > > uncountability of the reals > > No, it establishes the incompleteness of infinity or the infinity of > incompleteness.
And until WM can produce a surjection from N to R, none of his claims that the reals are countable show that the definition of countability is met. > > Cantor's list establishes the uncountability of distinguishable and > hence constructable reals. Why should nonconstructable and hence > nondistinguishable reals matter in Cantor's argument?
Until WM can produce a surjection from N to R, none of his claims that the reals are countable show that the definition of countability can be met. > > Cantor proves the uncountability of a countable set. For some people > that has an effect like a drug.
And until WM has produced a surjection from N to R, he is just blowing hot air. 

