```Date: Jan 4, 2013 1:47 PM
Author: fom
Subject: Re: The Distinguishability argument of the Reals.

On 1/4/2013 10:35 AM, WM wrote:> On 4 Jan., 13:36, 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.>> Therefore they cannot interfere with Cantor's argument and cannot> result from his procedure.Correct.  The diagonal argument is not the definitionof the reals.>>> 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.The latter of the two statements is a better choice.  The rationalsare not complete.  So much so, in fact, that they are a setof measure zero.But, wait.  A set of measure zero presumes a sigma algebra generatedfrom the open sets of the topology (or the compact sets if youprefer).>> Cantor's list establishes the uncountability of distinguishable and> hence constructable reals.Constructible real has a definite sense that youdo not abide by.  You should talk of nameable reals andFrege's notion of definite symbols.> Why should nonconstructable and hence> nondistinguishable reals matter in Cantor's argument?>> Cantor proves the uncountability of a countable set. For some people> that has an effect like a drug.Cantor proves that names isolated from thesystemic relation of their definition aresubject to local finiteness conditions.
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