Re: Undecidability

On Sun, 27 Nov 2005 04:15:34 +1100, Peter Webb wrote:

> "Herman Rubin" <hrubin@xxxxxxxxxxxxxxxxxxxx> wrote in message
> news:dm7do7$4p7c@xxxxxxxxxxxxxxxxxxxxxxx
>> In article <438734b4$0$8739$afc38c87@xxxxxxxxxxxxxxxxxxxx>,
>> Peter Webb <webbfamily-diespamdie@xxxxxxxxxxxxxxx> wrote:
>>>For many collections of sets, a choice function
>>>> can be explicitly found.
>>
>>>I don't think this is true. The only infinite set for which ANY choice
>>>function is known is N.
>>
>> This is not the case. Every well-ordered set has the
>> property that every non-empty subset has a smallest
>> element. One can give an algorithm to choose an element
>> from any non-empty bounded closed subset of the reals, or
>> from any open set in a topological space with a given
>> countable dense set. If one has a closed strictly convex
>> set in a locally convex metric metric space, one can
>> define a choice element.
>>
>>

> I am clearly out of my depth here.

> Here is my logic.

> Assume CH..

> Pick any set F. Any set of cardinality greater than Aleph0 must contain a
> set of the same cardinality as R as a subset, as the cardinals are well
> ordered. But we know of no explicit choice function over R. It seems to me
> (!) that if we had a choice function for a set of cardinality greater than
> R, we could use this to form a choice function over R by considering the
> choice as only being over R (a subset of F). So the only sets with a choice
> function have cardinaility less than R, meaning Aleph0.

I'm not sure what you mean by "a choice function on R", since the concept
of a choice function applies to a set of sets, and R is a set of numbers.
Are you viewing the members of R as Dedekind cuts, perhaps?

Let F be the set of all subsets of the rationals that are nonempty,
bounded above, downward closed, and have no maximum element. This is
essentially the set of all Dedekind cuts, except that I am considering
only the left sets of the cuts.

Then F has the same cardinality as R. There is clearly a choice function
on F. For example, for each A in F I can take the greatest integer in A.

Does this help you to construct a well-ordering on R? If so, how?





--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>
.

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