# 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