# [FOM] Re: Definition of Large Cardinal Axiom

Ali Enayat enayat at american.edu
Sat Apr 17 15:27:29 EDT 2004

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Professor Solovay caught my blooper in my recent e-mail (reproduced
below),in which I used "Baire Property" instead of "Lebesgue measurable".

To compensate for my carelessness, I looked up Shelah's paper to answer
Solovay's query whether Shelah needed DC (dependent choice)in his proof. It
turns out Shelah's proof needs only *countable choice*,i.e.,

Theorem A (Shelah) Con(ZF + "Every set of reals is Lebesgue measurable" +
"countable Choice") implies Con(ZFC + "There exists an inacessible
cardinal").

[Reference: "Can you take Solovay's inaccessible away?", Israel J. Math, 48
(1984), pp.1-47]

I should also point out that a much earlier example of a "regularity
property of reals" implying a large cardinal axiom (and I suspect the
first) is due to Specker (cf. p.135 of Kanamori's THE HIGHER INFINITE), who
established:

Theorem B (Specker, 1957) Con(ZF+ countable Choice + "Every uncountable set
of reals has a perfect subset") implies Con(ZFC + "There exists an
inacessible cardinal").

The converse of Theorem B was proved by Solovay, in the same celebrated
1970 paper - and in the same model - in which he also established the
converse of Theorem A (the main results of Solovay's paper were obtained in
1964).

Ali Enayat

>
>
>
> On Wed, 14 Apr 2004, Ali Enayat wrote:
> [snip]
> >
> > (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
> >
> > _______________________________________________
> > FOM mailing list
> > FOM at cs.nyu.edu
> > http://www.cs.nyu.edu/mailman/listinfo/fom
> >
>

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

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