NetNews Usenet Archive 1992 #16/NN_1992_16.iso/spool/sci/math/9391

Text File | 1992-07-21 | 3.7 KB | 81 lines

Newsgroups: sci.math
Path: sparky!uunet!caen!zaphod.mps.ohio-state.edu!magnus.acs.ohio-state.edu!wjcastre
From: [e-mail] (W.Jose Castrellon G.)
Subject: Re: Choice and Measurability
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Organization: The Ohio State University,Math.Dept.
References: <[e-mail]>
Date: Tue, 21 Jul 1992 23:38:59 GMT
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David G. Caraballo  writes
>In article  <[e-mail]> I wrote
>
>> Let M be the axiom  "All subsets of R^n are measurable".  Then the
>> following is true:
>> M is consistent with ZF and Countable Choice, but not with ZF and
>> AC.   Does someone have a reference?
>
>In article  <[e-mail]>  Allan Adler wrote
>
>> David Carabalo believes that Solovay proved that the nonexistence of
>> nonmeasurable sets is consistent with ZF+DC.
>
>For the record, David Caraballo believes no such thing.  The assumption
>that there exists an inaccessible cardinal is essential here, as Shelah
>proved.  I wrote "Countable Choice", not DC ("Dependent Choice").  I believe
>it was Bernays (Is this the correct attribution?) who proved that
>AC => DC => Countable Choice.  Consistency with ZF+DC is, of course, much
>more useful (DC allows us to prove all sorts of useful results, such as
>"The union of a countable number of countable sets is countable.").
>
>The relevant result that actually appears in Solovay's paper mentions
>DC, not Countable Choice.  The complete reference, by the way, is
>Annals of Math, 92; 1970, pp. 1-56.   However, consider the following
>reference which someone sent to me in response to my post:
>
>"Solovay in 1964 proved that the assertion 'all sets are Lebesgue
>measurable' is consistent with ZF and a restricted version of choice.
>This restricted version of choice, called the countable axiom of choice,
>asserts that every countable set of non-empty sets has a choice function."
>                         Malitz, _Introduction to Mathematical Logic_, p.49
>
>I had something similar in my own notes (which is why I posted something to
>this effect -- included above).  I am not prepared to disregard my notes
>(and now, the above reference) without seeing an actual proof that my claim
>"M is consistent with ZF and Countable Choice" is false.  If someone has a
>proof, I would love to see it.  Thank you.
>
>David G. Caraballo
>

I'm reluctantly writing this; I e-mailed yesterday one of the contributors to
the thread and he thought I should post it, some fellow students also urged me
to do it.
I dont wish to be involved in the controversy, so if anybody disagrees with
what follows or finds it inaccurate, please contact an expert that knows about
these things, or write directly to S.Shelah (Institute of Mathematics. The
Hebrew University of Jerusalem. Jerusalem, Israel).

"Can You Take Solovay's Inaccesible Away?" By Saharon Shelah. Israel Journal of
Mathematics, p.1-47. Vol. 48, No. 1, 1984.
p.18 reads:
" Main Theorem.  If every sigma^1 _3 set of reals is measurable,
then Aleph_1 is an inaccessible cardinal in L.
Remarks.
(1) The theorem is proved in ZFC, of course. However, very little use of the
axiom of choice is made, only that Aleph_1 is not singular in any L[a], a
a real. For this it suffices that Aleph_1 is regular, which follows from the
  **countable axiom of choice**   (i.e. the existence of choice for a family of
countably many sets). "

[the ** are mine].
Hopefully this will bring the controversy to an end.
W.Jose Castrellon G.
Grad mailroom.Math.Dept.
The Ohio State University
[e-mail]