Re: Zorn's lemma for families of subsets of a countable set

From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 06/24/04 

Date: Thu, 24 Jun 2004 05:27:26 -0500

On 23 Jun 2004 22:03:48 -0700, w.taylor@math.canterbury.ac.nz (Bill
Taylor) wrote:

>kramsay@aol.com (Keith Ramsay) wrote:
>
>> Is the special case of Zorn's lemma where the ordered
>> set consists of a family of subsets of a countable set,
>> ordered by inclusion, a theorem of ZF?
>
>This doesn't quite parse properly, to me, though others seem to have had
>no trouble. I presume it means:-
>
>____________________________________________________________________
>There is a countable set, C, and we wish to well-order (some or all)
>subsets of it. Is it a ZF-theorem that this can always be done?
>""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""

No, it means the following. (It's clear that a yes to your
question would imply a yes to the following, but it's not
so clear to me that the converse holds):

Can we prove in ZF that if C is a subset of the power
set of N, ordered by inclusion, and every chain in C has an
upper bound in C, then C has a maximal element?

>The answer is obviously "no". WLOG, the countable set can be taken to be N,
>(unless it's finite, making the result trivially true); then the set of ALL
>subsets of this is just R, and cannot be well-ordered in ZF, as is well-known.
>
>This seems so simple that I presume the original enquiry meant something else?
>
>Oh hang on, I see it did - Zorn's lemma *starts off* with an ordered set
>of a particular sort, doesn't it. Damn. But I assume my argument will
>go through largely as is.
>
>------------------------------------------------------------------------------
> Bill Taylor W.Taylor@math.canterbury.ac.nz
>------------------------------------------------------------------------------
> Set theory is a shotgun marriage - between well-ordering and power-set.
> The two parties get along OK; but they hardly seem made for each other.
>------------------------------------------------------------------------------

************************

David C. Ullrich

Relevant Pages