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 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.