December 10
Eyal Kaplan,
Tel Aviv University
Non-stationary support iterations of Prikry forcings and restrictions of ultrapower embeddings to the ground model: Part II
Assume that $\mathbb P$ is a forcing notion, $G$ is a generic set for it over the ground model $V$, and a cardinal $\kappa$ is measurable in the generic extension. Let $j$ be an ultrapower embedding, taken in $V[G]$ with a normal measure on $\kappa$. We consider the following questions:
1. Is the restriction of $j$ to $V$ an iterated ultrapower of $V$ (by its measures or extenders)?
2. Is the restriction of $j$ to $V$ definable in $V$?
By a work of Schindler [1], the answer to the first question is affirmative, assuming that there is no inner model with a Woodin Cardinal and $V=K$ is the core model. By a work of Hamkins [2], the answer to the second question is positive for forcing notions which admit a Gap below $\kappa$.
We will address the above questions in the context of nonstationary-support iteration of Prikry forcings below a measurable cardinal $\kappa$. Assuming GCH only in the ground model, we provide a positive answer for the first question, and describe in detail the structure of $j$ restricted to $V$ as an iteration of $V$. The answer to the second question may go either way, depending on the choice of the measures used in the Prikry forcings along the iteration; we will provide a simple sufficient condition for the positive answer. This is a joint work with Moti Gitik.
[1] Ralf Schindler. Iterates of the core model. Journal of Symbolic Logic, pages 241–251, 2006.
[2] Joel David Hamkins. Gap forcing. Israel Journal of Mathematics, 125(1):237–252, 2001.