Spencinar. The Spencinar is a graduate student seminar started and organized last year by Spencer Unger (blame Zach Norwood for the corny name). In this seminar, graduate students and Spencer meet for two hours weekly to present interesting results that they've read about and want to share, or little research ideas which are not appropriate for the Cabal Seminar. It's one way we can help each other acquire knowledge more quickly, and a great excuse for grabbing a drink afterwards with our fellow set theorists!
These posts are based off of my notes taken during the seminar, and any mistakes herein should not be attributed to the speaker. You should expect the notes to be less thorough in the future, as they are based on my own comprehension of the lecture, which can be extremely lacking outside of my immediate research interests.
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This year, Spencer Unger started off with Solovay splitting: Smash it with a hammer.
Prerequisites: Knowledge of ultrapowers and forcing, some club guessing results will be stated as black boxes in the second part.
Today's Spencinar was about saturated ideals. The title of the talk refers to Solovay's theorem about splitting stationary sets, and the hammer is forcing. It leads to a proof of the theorem (in fact, the original one) that seems more principled than the elementary combinatorial one in Jech's book, but Spencer notes that the essence of the two proofs is the same.
Theorem (Solovay splitting): If $S\subseteq \kappa$ is stationary, then there $\langle S_i: i<\kappa\rangle$ disjoint stationary subsets of $S$.
Recall that an ideal $I$ on $X$ is $\kappa$-saturated if $P(X)/I$ is $\kappa$-c.c. Here $P(X)/I$ is the poset whose elements are $\mod I$-equivalence classes of subsets of $X$, without the equivalence class consisting of the members of $I$. The ordering is by $[A] \leq [B]$ iff $A \setminus B \in I$ (i.e., inclusion $\mod I$). An equivalent formulation of $\kappa$-saturation is that for any $\kappa$ many $I$-positive subsets of $X$, there are two which intersect in an $I$-positive set.
Let us denote the nonstationary ideal on $\kappa$ by $NS_\kappa$. If $S$ is stationary, then $NS_\kappa \upharpoonright S$ is the ideal generated over $NS_\kappa$ by adding $S$.
Lemma: If Solovay splitting fails for $S$, then $NS_\kappa\upharpoonright S$ is $\kappa$-saturated.
Proof: An easy "disjointify" argument using $\kappa$-completeness. Let $\langle A_\alpha : \alpha <\kappa\rangle$ be stationary such that $A_\alpha \cap A_\beta \in NS_\kappa\upharpoonright S$ for all $\alpha \neq \beta$. Define $B_\alpha = A_\alpha - \bigcup_{\beta<\alpha} A_\beta$. Then $\langle B_\alpha: \alpha<\kappa\rangle$ witnesses Solovay splitting for $S$. $\Box$
Lemma (*): If $S\subseteq \kappa \cap \mathrm{cof}(>\omega)$ is stationary, then $T=\{\alpha\in S: S\cap \alpha \textrm{ is nonstationary}\}$ is stationary. (Note: $T$ is the set of points in $S$ that do not reflect $S$).
Proof: Fix a club $C\subseteq \kappa$. Let $C'$ be the set of limit points of $C$. If $\alpha$ is the least element in $S\cap C'$, then $\alpha$ does not reflect $S$ (as $C\cap \alpha$ is club in $\alpha$), so $\alpha \in T$. Since $C$ was an arbitrary club, the lemma is proved. $\Box$
Now we are ready to prove the Solovay splitting theorem.
Suppose the result fails for some $S$. Split into cases (Spencer hesitated about this, he really wanted a uniform proof).
Case 1. If $S$ concentrates on $\mathrm{cof}(\theta)$ for some $\theta<\kappa$, then forcing with $P(\kappa)/NS_\kappa\upharpoonright S\cap \mathrm{cof}(\theta)$ gives a $V$-ultrafilter $U$ concentrating on $S\cap \mathrm{cof}(\theta)$ which is $V$-$\kappa$-complete. This means the ultrafilter measures sets from $V$, and is closed under intersections of families in $V$ of $<\kappa$ many measure one sets, with the tacit meaning that the ultrafilter itself is not in $V$.
In the generic extension, form the ultrapower $j:V\rightarrow \mathrm{Ult}(V,U)$. Then $\mathrm{Ult}(V,U) \vDash \kappa \in j(S)$ (this was glossed over and uses the fact that $U$ is normal and $V$-$\kappa$-complete, which follows from a genericity argument), so $M \vDash \mathrm{cf}(\kappa)=\theta$. This is a contradiction.
Case 2. The remaining case is that $S$ concentrates on $\{\alpha : \mathrm{cf}(\alpha)=\alpha\}$. Let $T\subseteq S$ be defined as in Lemma (*). Since $T$ is stationary, it is an element of the forcing $P(\kappa)/NS_\kappa\upharpoonright S\cap \mathrm{cof}(\theta)$ so force below $T$ to get a generic $U$. Taking the ultrapower by $U$, $M\vDash \kappa\in j(T)$, so $$M\vDash j(S)\cap \kappa \textrm{ is not stationary. }$$
But $j(S)\cap \kappa=S$ and $\kappa$-c.c. forcing preserves stationarity of subsets of $\kappa$, so we again arrive at a contradiction.
Thus Solovay's theorem is proved.
I'll type up the second part of the talk later, which investigated the possibility of the nonstationary ideal on $\kappa$ being $\kappa^+$-saturated.
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