Dear Sy,
Theorem. Assume PD. Then there is a countable ordinal and a real
such that if M is a ctm such that
(1) is in M and
(2) M satisfies (this
but allowing
as a parameter),
then M is #-generated.
So, you still have not really addressed the ctm issue at all.
Here is the critical question:
Key Question: Can there exist a ctm M such that M satisfies in the hyper-universe of
where
is
-generic for collapsing all sets to be countable.
Or even:
Lesser Key Question: Suppose that M is a ctm which satisfies . Must M be #-generated?
Until one can show the answer is “yes” for the Key Question, there has been no genuine reduction of this version of to
-logic.
If the answer to the Lessor Key Question is “yes” then there is no possible reduction to -logic.
The theorem stated above strongly suggests the answer to the Lesser Key Question is actually “yes” if one restricts to models satisfying “”.
The point of course is that if M is a ctm which satisfies “” and M witnesses
then
witnesses
where
is an
-generic collapse of
to $lateex \omega$.
The simple consistency proofs of Original- all easily give models which satisfy “
”.
The problem
(*) Suppose
is not correctly computed by HOD for any infinite cardinal
. Must weak square hold at some singular strong limit cardinal?
actually grew out of my recent AIM visit with Cummings, Magidor, Rinot and Sinapova. We showed that the successor of a singular strong limit kappa of cof omega can be large in HOD, and I started asking about Weak Square. It holds at kappa in our model.
Assuming the Axiom is consistent one gets a model of ZFC in which for some singular strong limit
of uncountable cofinality, weak square fails at
and
is not correctly computed by HOD.
So one cannot focus on cofinality (unless Axiom
is inconsistent).
So born of this thread is the correct version of the problem:
Problem: Suppose is a singular strong limit cardinal of uncountable cardinality such that \gamma^+ is not correctly computed by HOD. Must weak square hold at \gamma?
Aside: asserts there is an elementary embedding
with critical point below
.
Regards, Hugh