Re: Paper and slides on indefiniteness of CH

Dear Sy,

Theorem. Assume PD. Then there is a countable ordinal \eta and a real x such that if M is a ctm such that

(1) x is in M and M \vDash ``V = L[t] \text{ for a real }t"

(2) M satisfies (*)(\eta) (this (*) but allowing \eta 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 L(M)[G] where G is L(M)-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 \textsf{IMH}^\# to V-logic.

If the answer to the Lessor Key Question is “yes” then there is no possible reduction to V-logic.

The theorem stated above strongly suggests the answer to the Lesser Key Question is actually “yes” if one restricts to models satisfying “V = L[Y]\text{ for some set }Y”.

The point of course is that if M is a ctm which satisfies “V = L[Y]\text{ for some set }Y” and M witnesses (*) then M[g] witnesses (*) where g is an M-generic collapse of Y to $lateex \omega$.

The simple consistency proofs of Original-\textsf{IMH}^\# all easily give models which satisfy “V = L[Y]\text{ for some set }Y”.

The problem

(*) Suppose \gamma^+ is not correctly computed by HOD for any infinite cardinal \gamma. 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 \textsf{I}0^\# is consistent one gets a model of ZFC in which for some singular strong limit \kappa of uncountable cofinality, weak square fails at \kappa and \kappa^+ is not correctly computed by HOD.

So one cannot focus on cofinality \omega (unless Axiom \textsf{I}0^\# is inconsistent).

So born of this thread is the correct version of the problem:

Problem: Suppose \gamma 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: \textsf{I}0^\# asserts there is an elementary embedding j:L(V_{\lambda+1}^\#) \to L(V_{\lambda+1}^\#) with critical point below \lambda.

Regards, Hugh

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