[FOM] Re: Definition of "large cardinal axiom"?

Robert M. Solovay solovay at math.berkeley.edu
Wed Apr 14 23:33:18 EDT 2004

On Wed, 14 Apr 2004, Ali Enayat wrote:
> (3) Some mathematical statements might *imply* a large cardinal axiom as in
> (1), or they might imply the truth of a large cardinal axiom in some inner
> model of set theory (such as Godel's constriuctible universe L). Often such
> statements are also referred to as a large cardinal axiom. For example, the
> statement "all subsets of reals have the property of Baire" is known to
> imply that "there is an inaccessible cardinal in L" (thanks to a theorem of
> Shelah in 1980).

	This is not correct. Shelah proved

	(a) Con(ZFC) iff Con(ZF + "All sets have the property of Baire");

        (b) Con(ZF + "Every set of reals is Lebesgue measurable" + DC)
implies Con(ZFC + "There exists an inacessible cardinal"). The converse
direction had been proved some years earlier by me.

	I'm not sure whether or not Shelah needed DC in (b).

	--Bob Solovay

> Ali Enayat
> Department of Mathematics and Statistics
> American University
> 4400 Massachusetts Ave, NW
>  Washington, DC 20016-8050
> (202) 885-3168
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