fom
Posts:
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Registered:
12/4/12


Re: The Distinguishability argument of the Reals.
Posted:
Jan 4, 2013 1:47 PM


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 >>> <a60601d524a24501a28b84a7b1e53...@ci3g2000vbb.googlegroups.com>, >>> WM <mueck...@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. > > 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.

