Dear Sy,
On Nov 3, 2014, at 3:38 AM, Sy David Friedman wrote:
Hugh:
1. The only method I know to obtain the consistency of the maximality criterion I stated involves Prikry-like forcings, which add Weak Squares. Weak Squares contradict supercompactness.
So you think that if the Maximality Criterion holds then weak square holds at some singular strong limit?
3. I was postponing the discussion of the reduction of #-generation to ctm’s (countable transitive models) as long as possible as it is quite technical, but as you raised it again I’ll deal with it now. Recall that in the HP “thickenings” are dealt with via theories. So #-generation really means that for each Gödel lengthening
of
, the theory in
-iterable is consistent. Another way to say this is that for each
in which
is (weakly) #-generated if for each countable
as a statement about ctm’s. In conclusion: You are right, we can’t apply LS to the raw version of
property; but weak #-generation is
and this is the only change required.
Just be clear you are now proposing that
1)
2) If
holds in an outer model of
Here: a ctm
is weakly #-generated if for each countable ordinal
whose
-iterate gives
Is this correct?
Regards, Hugh
of 
, the theory in 
-iterable is consistent. Another way to say this is that for each 
in which 
is (weakly) #-generated if for each countable 
as a statement about ctm’s. In conclusion: You are right, we can’t apply LS to the raw version of 
property; but weak #-generation is 
and this is the only change required.
holds in an outer model of 
is weakly #-generated if for each countable ordinal 
whose 
-iterate gives