[FOM] Re: Why the definition of "large cardinal axiom" matters

[Roger Bishop Jones]

On Tuesday 20 April 2004  5:53 pm, Timothy Y. Chow wrote:
> Roger Bishop Jones wrote:
> >Should we expect large cardinals to settle CH, or should
> >we suspect that a large cardinal which does so is saying
> >more than a large cardinal axiom should?
>
> Levy-Solovay and Cohen showed that large cardinals cannot be
> expected to settle CH.
>
> Or so I'm told...I don't understand their work myself.  Could
> someone explain these results in intuitive terms?  (I
> understand how to show that CH is independent of ZFC.)

I noticed this point in Woodin's lectures just after posting
my question.

In relation to my previous question this suggests that they
must have had a definition of "large cardinal axiom" which
is stronger than the definition mooted by Joe Shipman, 
for I see nothing Joe's definition which would prevent at
least one of the following from being a large cardinal axiom:

  exists kappa st (kappa is inaccessible and CH)
or
  exists kappa st (kappa is inaccessible and not CH)

(or something equivalent or stronger but less blatant)

Woodin does say something like "subject to modest conditions",
so the extra you need might be in those conditions rather
than in the definition of large cardinal axiom, but either
way I would love to know exactly what the conditions were
which permitted the derivation of this result.

I wonder if anyone is able to expand on Woodin's reference
which was just "(Levy Solovay, 1964; Cohen, 1965 )"?

Roger Jones

- rbj01 at rbjones.com
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