Following Larson's "The Stationary Tower", let $\mathbb{P}_{<\delta}$ be the full stationary tower on $\delta$, and for a stationary $S\subset \mathcal{P}_\delta(V_\delta)$, $\mathbb{P}_{<\delta}^S$ is the restriction to $S$.

It is stated that "under fairly general assumptions on $S$, much of the basic theory of $\mathbb{P}_{<\delta}$ carries over to $\mathbb{P}_{<\delta}^S$", but the book then deals only with the case $S=\mathcal{P}_{\omega_1}(V_\delta)$. My questions are:

  1. What are the general assumptions? Is there a reference for the general case?
  2. More specifically, if we let $S=\mathcal{P}_{\kappa}(V_\delta)$ for $\kappa<\delta$, will the same theorems still apply? I suppose we need $\kappa$ regular, does it need to be successor cardinal?
  3. When $S=\mathcal{P}_{\omega_1}(V_\delta)$ and $\delta$ is Woodin, we get in the extension an embedding $j$ such that $crit(j)=\omega_1$ and $j(\omega_1)=\delta$. If $S=\mathcal{P}_{\kappa}(V_\delta)$, will we get $crit(j)=\kappa$ and $j(\kappa)=\delta$? If not, how (and for which $\kappa$) can we get such an embedding?
  • $\begingroup$ I sent you Menachem's lecture notes, right? $\endgroup$ – Asaf Karagila Jan 16 '18 at 17:37
  • $\begingroup$ Yes but I think he also did only the full and the countable cases, no? $\endgroup$ – Ur Ya'ar Jan 17 '18 at 9:09
  • $\begingroup$ Hm. Yes, you're right. I misread your question... (also, note that Menachem replaced $\Bbb P$ and $\Bbb Q$ in his notes, and much to his dismay at the end of the year he said that he tried to be consistent with Paul's notation... :)) $\endgroup$ – Asaf Karagila Jan 17 '18 at 9:41

Here is a partial answer. Ideally, I'll think about it a little more and edit it.

Remark 2.7.17 of my book gives some pointers in the direction of your questions, and the note "Six lectures on the stationary tower" on my webpage gives more information about the case $S = \mathcal{P}_{\kappa}(V_{\delta})$, for $\kappa$ a succesor cardinal. In both cases I leave a lot to the reader. I think everything in the six lectures applies to regular cardinals; I just didn't write it that way. I don't think there's a reference for the general case, but the Foreman-Magidor paper on definable counterexamples to the continuum hypothesis covers some cases that I didn't, like the $\omega$-closed tower.

As for your third question, if $\kappa$ is a regular cardinal, then yes, $\kappa$ is the critical point of the embedding, and $j(\kappa) = \delta$. I think all the standard arguments carry over in this case. Even if $\kappa$ is singular, as I recall, you get an ultrapower which is closed under $\omega$-sequences, and possibly longer sequences, depending on the generic filter, but with critical point less than $\kappa$. I think this is done in one of the papers that I point to in Remark 2.7.17.

As for the question about arbitrary $S$, I suppose that one would like to characterize the ones that give you wellfounded ultrapowers. This might be hard, but maybe one can prove something generalizing the situation for the $\mathcal{P}_{\kappa}(V_{\delta})$'s. To start with, one would like to know what you need to prove Lemma 2.5.6 and Theorem 2.5.9 of my book, but even then this might not give you a characterization.

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