|
|
Re: The Distinguishability argument of the Reals.
Posted:
Jan 4, 2013 11:08 PM
|
|
On Jan 4, 4:36 am, gus gassmann <g...@nospam.com> wrote:
> On 03/01/2013 5:53 PM, Virgil wrote:
>
>
>
>
>
>
>
>
>
> > In article
> > <a60601d5-24a2-4501-a28b-84a7b1e53...@ci3g2000vbb.googlegroups.com>,
> > WM <mueck...@rz.fh-augsburg.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. Any *one* number not on
> the list shows that the list is incomplete and thus establishes the
> uncountability of the reals.
No, they are assumed to have expansions and need not be written.
Consider the function that is the limit of functions f(n,d) = n/d, n =
0, ..., d; n, d E N. In binary, its expansions would only start .000
{m many zeros} 000..., the only antidiagonal is .111... = 1.0 =
f(d,d).
The antidiagonal is in the range of the function from the naturals, at
the end.
The reals there are distinguished from one another by their constant
monotone progression.
Regards,
Ross Finlayson
|
|
