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