(other relevant posts for reference: Spencinar #3: precipitous ideals)
Let's begin! Remember that one of our goals is
Theorem: $\rm{Col}(\mu,<\kappa)$ makes the nonstationary ideal restricted to $[\mu]^{<\mu}$ precipitous.
First, let's define notions of stationarity on sets of the form $[\lambda]^{<\kappa}=\{x\subseteq \lambda: |x|<\kappa\}$. We will have $\kappa$ regular.
There are two different notions here. If $X$ is a set and $C\subseteq P(X)$, then $C$ is strongly club if there is a structure $\mathfrak{A}$ on $X$ with countably many function symbols so that $C$ is the set of elementary substructures of $\mathfrak{A}$ (or equivalently, the set of substructures closed under a function $X^{<\omega}\rightarrow X$). $C$ is J-club if it is unbounded in the $\subseteq$ order on $[\lambda]^{<\kappa}$ and closed under increasing unions of size less than $\kappa$. It turns out the two notions are closely connected, since a result of Kueker says that the filter generated by the J-club sets is the filter generated by the strongly club sets together with all sets of the form $\{x\in [\lambda]^{<\kappa}:x\cap \kappa \in \kappa\}$ (this means that the ordinals of $x\cap \kappa$ are an initial segment of all ordinals).
We can correspondingly define weakly stationary (resp. J-stationary) sets as those that intersect every strongly club (resp. J-club) set. For our purposes, strongly club and weakly stationary sets are the right notions, and we will refer to them from now on without the adverbs.
In this post, we will get a simple criterion for precipitousness. Read the Spencinar #1 or #3 posts for basic information about generic ultrapowers.
Our notation will be that $Z$ is a set and $I$ is an ideal on $Z$. We will use $I^+$ to denote the collection of $I$-positive sets, i.e., the set $P(Z)-I$. We will use $I^\wedge$ to mean the dual filter of $I$, i.e., the collection of sets of the form $Z-A$ for some $A\in I$. The equivalence class of $S\subseteq Z$ in $P(Z)/I$ will be denoted by $[S]$.
Definition: A tree of maximal antichains $\mathcal{A}$ for $P(Z)/I$ below $S$ is an underlying tree $T\subseteq \mathrm{ON}^{<\omega}$ labeled with $\langle A_s: s\in T\rangle$ elements of $P(Z)$ such that
- $A_\emptyset = S$,
- The levels of the tree form increasingly refined maximal antichains of $P(Z)/I$ restricted to $S$, i.e., for each $s\in T$, the set $\{[A_{s\ast \alpha}]:s\ast\alpha\in T\}$ is a maximal antichain below $A_s$ ($\ast$ denotes concatenation). Further, let us assume that $A_{s\ast \alpha}\subseteq A_s$. Denote the levels by $\mathcal{A}_n$, $n<\omega$.
Proof: If $I$ is precipitous, and $\langle A_s: s\in T$ is a tree of maximal antichains below some $S\in P(Z)$, then consider the generic ultrapower (see older posts for references) by $P(Z)/I$ restricted to $[S]$. This is a map
$$j:V\rightarrow M\subseteq V[G]$$
defined in the forcing extension by $G$ which is generic for $P(Z)/I$ restricted to $[S]$. We will think of $G$ as a $V$-ultrafilter extending the dual filter of $I$.
Let's review the construction of this ultrapower. The elements are represented by functions $F: Z\rightarrow V$ in $V$, and we have Los's theorem which says for a formula $\varphi$ in the language of set theory and $F_0,\ldots,F_n$, $M\vDash \varphi([F_0],\ldots,[F_n])$ if and only if $\{z:V\vDash \varphi(F_0(z),\ldots,F_n(z))\}\in G$.
In $M$, let $\mathcal{B}=j(\mathcal{A})$ and $i$ be the element represented in the ultrapower by the identity function on $Z$.
For each $n<\omega$, there must be a unique $A_{s_n}\in \mathcal{A}_n$ so that $i\in j(A_{s_n})$: this $A_s$ is the one in $G$ (such $A_s$ must exist by a basic density argument using the fact that $\mathcal{A}_n$ is a maximal antichain, and it must be unique). The sequence of these $\langle A_{s_n}:n<\omega\rangle$ can be computed in $V[G]$ and is a branch through $\mathcal{B}$. Then there must be a branch in $M$ through the subtree of $\mathcal{B}$ consisting of those nodes which are labeled by a set containing $i$ as an element. (Otherwise, in $M$ there would be a rank function $f$ on this subtree taking ordinal values such that $f(B_s)<f(B_t)$ if $s$ is below $t$, and this would have to be in $V[G]$, so there couldn't be a branch in $V[G]$. Note that this is an important pattern of argument that is very common in many different places in set theory).
Now this branch is a branch through $\mathcal{B}$ with nonempty intersection (containing $i$, for one) so by elementarity there is such a branch through $\mathcal{A}$.
Conversely, assume that $I$ is not precipitous. Then there is $S\subseteq Z$ such that $[S]$ forces that the generic ultrapower is well-founded. By maximality of the forcing language, we can find names $\dot{F}_n$ such that
- $[S]\Vdash \dot{F}_n:Z\rightarrow V$,
- $[S]\Vdash \dot{F}_n\in V$,
- $[S]\Vdash ``M \vDash [\dot{F}_{n+1}]\in [\dot{F}_n]''$.
We can build a tree of maximal antichains $\mathcal{A}$ below $[S]$ such that
- a condition $A_s$ on the $n$th level decides the value of $\dot{F}_n$, say as $f^s_n$,
- if $s$ is the predecessor of $t$ in the underlying tree, then $f^t_{n+1}(z)\in f^s_n(z)$ for all $z\in A_t$.
This tree has no branch with nonempty intersection, otherwise this would give an infinite descending sequence in $V$. $\Box$
Another thing we will need is the notion of an internally approachable (IA) structure. Let $\lambda$ be sufficiently large regular cardinal and consider the structure $\mathfrak{A}=\langle H(\lambda),\epsilon,\Delta,f_i\rangle$ where the $f_i$ are countably many function symbols and $\Delta$ is a fixed well-order of $H(\lambda)$ needed for certain Skolem hull arguments. An elementary substructure $N\prec \mathfrak{A}$ is internally approachable iff it can be written as $N=\bigcup_{j<\delta}N_\delta$ for some $\delta$ such that for all $\beta\in \delta$, $\langle N_\alpha:\alpha<\beta\rangle\in N$ (this definition is slightly weaker than other common definitions of IA structures). Sometimes such an $N$ will be said to be internally approachable of length $\delta$.
Because of the way these structures are built in layers, an IA structure can do some pretty "meta" computations (as we will see). This enables us to do some inductive constructions, where the IA condition gives a kind of strengthened inductive hypothesis.
First we have a basic lemma about IA structures which is useful in many contexts.
Lemma 2: Let $N\prec \langle H(\lambda),\epsilon,\Delta$ be IA of length $\delta$ for $\delta<\lambda$. Suppose $a\in y$ for some $y\in N$. Then the Skolem hull of $N\cup\{a\}$ computed in $H(\lambda)$ is IA of length $\delta$.
Proof: Let $\langle N_\alpha:\alpha<\delta\rangle$ witness approachability. For each $\alpha<\delta$, let $N^*_\alpha$ be the closure of $N_\alpha\cup\{a\}$ under all functions $H(\lambda)\rightarrow H(\lambda)$ in $N_{\alpha+1}$. The $N^*_\alpha$ are elementary substructures of $H(\lambda)$ that witness approachability for the Skolem hull of $N\cup\{a\}$: if $\beta<\delta$, then $\langle N^*_\alpha:\alpha<\beta\rangle$ can be computed from $a$ and the sequence $\langle N_\alpha:\alpha\le \beta\rangle$. $\Box$
Next is the main importance of IA structures in this project. It is very typical of applications of approachability:
Lemma 3: Suppose $S\subseteq \rm{IA}$ be a stationary subset of $[H(\lambda)]^{<\mu}$, for some uncountable regular $\mu<\delta$. Then $S$ remains stationary in $[H(\lambda)^V]^{<\mu}$ after forcing with $\mathbb{P}=\rm{Col}(\mu,<\kappa)$ for any ordinal $\kappa$.
Proof: Let $\dot{F}$ be a name for a function $H(\lambda)^{<\omega}\rightarrow H(\lambda)$ in the extension. Let $\lambda^*$ be a sufficiently large regular cardinal, and $N\prec \langle H(\lambda^*),\in,\Delta,\mathbb{P},\dot{F},S\rangle$ with $|N<\mu|$ and $N\cap H(\lambda)\in S$.
Then $N\cap H(\lambda)$ is IA of some length $\delta<\mu$, witnessed by $\langle N_\alpha:\alpha<\delta\rangle$. We will define a decreasing sequence $\langle p_\alpha:\alpha<\delta\rangle$ of conditions in $\mathbb{P}$ so that
- For every $\beta<\delta$, there is $M_\beta\in N$ containing $N_\beta$ and each of the previous $M_\alpha$, $\alpha<\beta$, and so that for each $x\in M_\alpha$, $p_\beta$ forces there is $y\in M_\beta$ with $F(x)=y$.
- For every $\beta<\delta$, $\langle p_\alpha:\alpha<\beta\rangle\in N$.
We just pick the $\Delta$-least $p_\beta$ so that there is a $M_\beta$ satisfying the above, and then pick the $\Delta$-least such $M_\beta$. These things all exist in $N$ since $N$ is IA, and we can get (2) since this computation can be done inside $N$.
Now use closure of $\mathbb{P}$ to find a $p$ extending all of the $p_\alpha$. This $p$ forces that $N\cap H(\lambda)$ is closed under $F$, so $S$ is stationary.
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