fom
Posts:
1,099
Registered:
12/4/12
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Re: The Distinguishability argument of the Reals.
Posted:
Jan 4, 2013 1:47 PM
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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 definition
of 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 rationals
are not complete. So much so, in fact, that they are a set
of measure zero.
But, wait. A set of measure zero presumes a sigma algebra generated
from the open sets of the topology (or the compact sets if you
prefer).
>
> Cantor's list establishes the uncountability of distinguishable and
> hence constructable reals.
Constructible real has a definite sense that you
do not abide by. You should talk of nameable reals and
Frege'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 the
systemic relation of their definition are
subject to local finiteness conditions.
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