
Re: The Distinguishability argument of the Reals.
Posted:
Jan 4, 2013 7:36 AM


On 03/01/2013 5:53 PM, Virgil wrote: > In article > <a60601d524a24501a28b84a7b1e53bac@ci3g2000vbb.googlegroups.com>, > WM <mueckenh@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. Any *one* number not on the list shows that the list is incomplete and thus establishes the uncountability of the reals.

