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

From: Bill Taylor (w.taylor_at_math.canterbury.ac.nz)
Date: 06/24/04 

Date: 23 Jun 2004 22:03:48 -0700

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?
""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""

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
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    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.
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