Example 2.4.2 of Larson's The stationary tower (based on Woodin's lecture) describes how we can absorb a generic filter over a set forcing in a definable inner model into a generic extension by $\mathbb{P}_\infty$. Especially, it says the following holds:
Theorem. Let $K$ be the core model and suppose that it has a proper class of completely Jónsson cardinals. If $g$ is set generic over $K$, then there is a $\mathbb{P}_\infty^K$-generic filter $G\subseteq\mathbb{P}_\infty^K$ such that $g\in K[G]$.
One major property of the forcing $\mathbb{P}_\infty$ under the existence of a proper class of completely Jónsson cardinals is that we have a generic elementary embedding $j\colon V\to V[G]$. Since the formula $V=K$ is first-order expressible (I suppose this is true, but I am unsure about it because I have not learned inner model theory), we have that $V[G]$ also believes a version of $V=K$: that is, $V[G]=K^{V[G]}$.
My questions are:
- Does it mean the core model $K$ is not absolute (unlike $L$) if there is a proper class of completely Jónsson cardinals in $K$?
- When do we have a proper class of completely Jónsson cardinals in $K$? For example, Does $K$ has a proper class of completely Jónsson cardinals if $V$ does? (I guess the paper Jónsson Cardinals, Erdős Cardinals, and the Core Model by Mitchell is related to this question.)
- If the answer to my first question is yes, are there any known better results about the failure of the absoluteness of $K$? (Like, the failure of the absoluteness under a weaker large cardinal hypothesis.)
(A cardinal $\kappa$ is completely Jónsson if for each $a\in \mathbb{P}_\kappa$ the set $$\{X\subseteq V_\kappa \mid X\cap (\cup a)\in a \land |X\cap\kappa|=\kappa\}$$ is stationary in $\mathcal{P}(V_\kappa)$. A completely Jónsson cardinal is Jónsson, and a Ramsey cardinal is completely Jónsson.)
