Spencinar flashback: Approachability and stationary reflection at $\aleph_\omega$

The Spencinar has been continuing as usual since the Thanksgiving break. Two weeks ago, Spencer talked about diamond principles. I'm working on these notes still. Last week, I talked about precipitousness of the nonstationary ideal (the subject of the learning project #1).

I'm about to retire this old seminar notebook, so I thought I would record here a Spencinar that I gave over the summer about the approachability property. This will also provide some useful background for future posts.

This material comes mostly from Eisworth's friendly handbook chapter, where I was originally inspired to pursue the line of research that I'm on. I'm going to focus on the following result of Shelah's (interpreted through Eisworth's chapter):

Main Theorem: If $\aleph_\omega$ is a strong limit, then $\rm{Refl}(\aleph_{\omega+1})$ implies the approachability property at $\aleph_\omega$.

This result is unexpected since approachability is a rather weak "non-reflection" principle. It's interesting since Magidor's model of $\rm{Refl}(\aleph_{\omega+1})$ satisfies approachability property at $\aleph_\omega$, and in fact uses this to show that stationarity is preserved after forcing. It's also interesting since the proof seems to use special properties of $\aleph_\omega$, giving an example of the phenomenon that this singular cardinal in particular seems to have a lot of ZFC structure. This is nice to understand, because there is an important open question asking whether the failure of singular cardinals hypothesis at $\aleph_\omega$ implies approachability (and this is consistently false for larger singular cardinals).

I. $\mathrm{AP}_\mu$: review.

Shelah defined the ideal $I[\lambda]$ for a regular cardinal $\lambda$ by:

$S\subseteq I[\lambda]$ iff there is

1. a sequence $\langle a_\alpha:\alpha<\lambda\rangle$ of bounded subsets of $\lambda$ and
2. $C\subseteq \lambda$ club,

 so that for every $\delta\in S\cap C$, there is $A_\delta\subseteq \delta$ unbounded of order-type $\rm{cf}(\delta)$ so that

• $\{A\cap \beta:\beta<\delta\}\subseteq \{a_\beta:\beta<\delta\}$ (or in words, all initial segments of $A_\delta$ are enumerated in $\bar{a}$ at a stage before $\delta$.

Such $\delta$ is said to be approachable with respect to $\bar{a}$. It can be shown (cf. Eisworth's handbook chapter) that $I[\lambda]$ is a normal ideal on $\lambda$.

For $\mu$ a singular cardinal, the approachability property $\rm{AP}_\mu$ is the statement $\mu^+\in I[\mu^+]$. (Some other authors denote this by $\rm{AP}_{\mu^+}$. Restating the definitions, this means there is $\bar{a}$ and a club of $\delta<\mu^+$ of points which are approachable with respect to $\bar{a}$.

2. Elementary substructures

We will define a $\lambda$-approximating sequence to be a $\subseteq$-increasing sequence $\mathcal{M}=\langle M_\alpha:\alpha<\lambda\rangle$ of elementary substructures of $(H(\theta); \in, \triangleleft)$ for sufficiently large regular $\theta$ so that $\lambda\in M_0$ and for all $\alpha$, $|M_\alpha|<\lambda$ and $M_\alpha\cap \lambda\in \lambda$. Most importantly, we require that $\langle M_\alpha:\alpha\le \beta\rangle\in M_{\beta+1}$. This is a slight strengthening of the concept of "IA sequence" which we saw in the learning project.

If $\mathcal{M}$ is a  $\lambda$-approximating sequence, then define $S[\mathcal{M}]$ to be the set of $\delta<\lambda$ such that $M_\delta\cap\lambda=\delta$, there is a cofinal $a\subseteq \delta$ of order-type $\rm{cf}(\delta)$ such that every inital segment of $a$ is in $M_\delta$.

This gives a useful characterization of approachability. $I[\lambda]$ is just ideal generated by nonstationary sets together with sets of the form $S[\mathcal{M}]$. For a given $\bar{a}$, let $\mathcal{M}$ be a $\lambda$-approximating sequence with $\bar{a}\in M_0$ and $E\subset\lambda$ be club so that $M_\alpha\cap\lambda=\alpha$ for all $\alpha\in E$. In this club, the set of points approachable with respect to $\bar{a}$ is contained in $S[\mathcal{M}]\cap E$.

Conversely, given a sequence $\mathcal{M}$, let $\bar{a}$ be the sequence of bounded subsets of $\lambda$ in $\bigcup_\alpha M_\alpha$, and $E$ be the club of $\delta$ so that $\{a_\alpha:\alpha<\delta\}$ is an enumeration of the bounded subsets of $M_\delta$.

3. Colorings

We can also rephrase notions of approachability in terms of colorings, given some extra assumptions. Here $S^\lambda_\kappa$ denotes the set of points of cofinality $\kappa$ below $\lambda$.

Theorem 1: Suppose $\kappa<\lambda$ be regular with $2^{<\kappa}<\lambda$. If $d:[\lambda]^2\rightarrow \omega$ and $\mathcal{M}$ is a $\lambda$-approximating sequence containing $\{\kappa, d\}$, then for every $\delta\in S[\mathcal{M}]\cap S^\lambda_\kappa$, there is a cofinal $H\subseteq \delta$ homogeneous for d.

Proof: If $\delta\in S[\mathcal{M}]\cap S^\lambda_\kappa$, let $\{\alpha_i:i<\kappa\}$ cofinal in $\delta$ such that $\{a_i:i<\zeta\}\in M_\delta$ for all $\zeta<\kappa$. By induction on $i<\kappa$, we can define $\epsilon_i,f_i$ so that:

1. $\epsilon_0=\alpha_0$.
2.  $f_i:i\rightarrow \theta$, $f_i(j)=d(\epsilon_j,\delta)$.
3. $\epsilon_i$ is the least $\alpha$ so that for all $j<i$, $\alpha>\alpha_j,\epsilon_j$ and $d(\epsilon_j,\alpha)=d(\epsilon_j,\delta)$ for all $j<i$, if such exists. The construction ends if no such exists.

This last condition says that the function $d(\epsilon_i,\epsilon_j)$ depends only on $i$, where $i<j$, and uses $\delta$ as a "reference point". Let $i^*$ be the length of the construction.

We show the construction goes up to $\delta$. By part of the definition of $S[\mathcal{M}]$, $M_\delta\cap\lambda=\delta$. Since $\kappa\in M_\delta$, $M_\delta$ must be $<\kappa$-closed, so it contains $f_i$ for all $i<i^*$ (this uses $2^{<\kappa}<\lambda$). This means the construction above can be done in $M_\delta$ and therefore all $\epsilon_i<\delta$, $\delta$ witnesses condition in 3.

The proof concludes by finding $\xi$ so that $\{i<\kappa:d(\epsilon_i,\delta)=\xi\}$ is unbounded, giving a homogeneous set. $\Box$

This motivates yet another conception of "approachability".

Definition: For $d:[\lambda]^2\rightarrow\chi$ (some $\chi$), define $S(d)$ to be the set of $\delta<\lambda$ for which there is a cofinal $H_\delta\subseteq \delta$ homogeneous for $d$, and $S^*(d)=\lambda-S(d)$.

So $S(d)$ is club for every $d$ if the approachability property holds at the predecessor of $\lambda$ (under the hypotheses of Theorem 1).

Actually, under those hypotheses, the approachability ideal is generated by a single set over the nonstationary ideal. We can see this result in a way by using colorings. It turns out there is a certain class of colorings whose members have "all of the complexity" of the general case.

Definition: $d:[\lambda]^2\rightarrow\omega$ is a coloring as above with $\lambda=\mu^+$. We say $d$ is normal if $\{\beta<\alpha:d(\beta,\alpha)<i\}|<\mu$ for all $i<\rm{cf}(\mu)$. We say $d$ is transitive if $d(\alpha,\gamma)\le \mathrm{max}\{d(\alpha,\beta),d(\beta,\gamma)\}$ for all $\alpha<\beta<\gamma<\lambda$.

Normal transitive colorings are not difficult (but a little tricky) to construct. We'll just take this as an unproven fact (this construction was used, for example, in John's Spencinar talk about Martin's Maximum).

Proposition: If $\mu$ is strong limit and $d$ is a normal, transitive coloring of $\mu^+$, then there is $\bar{a}$ so that almost every point of $S(d)$ is approachable with respect to $\bar{a}$.

Proof: Let $\langle a_\xi:\xi<\lambda\rangle$ enumerate all subsets of the form $\{\beta<\alpha:d(\beta,\alpha)<i\}$ in some order. This is fine since $\lambda^{<\lambda}=\lambda$ ($\mu$ strong limit). Then for any $\delta\in S(d)$ of uncountable cofinality there exists a cofinal $H_\delta\subseteq\delta$ homogeneous for $d$. Actually, $d(\beta,\delta)$ is bounded in $\omega$: by transitivity, if $\beta_1<\beta_2<\delta$, then $d(\beta_1,\delta)\le \max\{i,d(\beta_2,\delta)\}$, and since $\delta$ has uncountable cofinality and $d(\beta,\delta)$ is increasing in $\beta\in H_\delta$ (if above $i$), then $d(\beta,\delta)$ is bounded. So almost every point of $S(d)$ is approachable. $\Box$

So the proposition gives a really nice characterization of approachability property under the hypotheses of Theorem 1: $\mathrm{AP}_\mu$ holds iff there exists (equivalently for all) normal transitive colorings, $S(d)$ contains a club of points of uncountable cofinality.

4. Proof of Main Theorem

Now we start working towards the proof of the theorem. The first result is that $S^*(d)$ can only reflect in itself.

Lemma 2: If $S^*(d)\cap \alpha$ is stationary in $\alpha$, then $\alpha\in S^*(d)$.

Proof: Otherwise $\alpha\in S(d)$, so let $H\subset \alpha$ be cofinal in $\alpha$ and homogeneous for $d$. Then every limit point of $H$ must be in $S(d)$ by the definition of $S(d)$, so $S^*(d)$ is disjoint from a club in $\alpha$. $\Box$

So if the approachability property fails, then there is a coloring $d$ (namely, any normal transitive one) for which $S^*(d)\cap \mathrm{cof}(>\omega)$ is stationary. Take $n<\omega$ such that $S=S^*(d)\cap S^{\aleph_{\omega+1}}_{\aleph_{n+1}}$ is stationary.

The next claim is the crucial fact giving a non-reflecting stationary set.

Claim: $S$ cannot reflect at points of cofinality $>2^{\aleph_n}$.

Proof of Claim: Suppose $S$ reflects at $\tau$.  If $\tau>2^{\aleph_n}$, then $\tau^{<\aleph_{n+1}}=\tau$. So we can construct a $\lambda$-approximating sequence containing all the relevant parameters which is $<\aleph_{n+1}$ closed. Then for any $\delta\in S^\tau_{\aleph_{n+1}}$ with $M_\delta=\delta$, any $A\subseteq \delta$ of order-type ${\aleph_{n+1}}$ has all of its initial segments in $M_\delta$. This shows that $S^\tau_{\aleph_{n+1}}\in I[\tau]$. This is impossible by Theorem 1, which says almost all points below $\tau$ are in $S(d)$, so $S$ cannot reflect at $\tau$. $\Box$

Now fix $k<\omega$ so that $2^{\aleph_n}=\aleph_{n+k}$. Such exists by $\mu$ strong limit. Define $S_0=S$, and $S_{i+1}$ to be the reflection points of $S_i$. Observe that if $\delta\in S_i$, then $\mathrm{cf}(\delta)\ge \aleph_{n+1+i}$, since the set of points below an ordinal with cofinality less than that ordinal is club. This implies that $S_k$ is empty, so let $i^*<\omega$ be maximal so that $S_{i^*}$ is stationary. Then $S_{i^*}$ does not reflect stationarily often. Hence we have proven the main theorem.

5. Normal scales?

Let's try to apply some of these ideas. This section requires a basic knowledge of pcf theory. Let $\langle f_\alpha:\alpha<\mu^+\rangle$ be a scale.

The "difference function" of the scale is a transitive function $d:[\mu^+]^2\rightarrow\omega$ defined so that $d(\alpha,\beta)$ for $\alpha<\beta$ is the supremum of all $n$ so that $f_\alpha(n)\ge f_\beta(n)$. In light of the previous explorations on approachability, it is natural to ask if it can ever be normal.

Proposition: Normal scales don't exist.

Proof: Let $\vec{f}$ be a scale on $\langle \mu_i:i<\omega\rangle$. Pick $\gamma_1<\mu_1$ so that $B_1=\{\alpha:f_\alpha(1)<\gamma_1\}$ has size $\kappa^+$. Let $A_0$ be a set of $\mu_0$ many $\alpha<\mu^+$ in $B_1$.

If $n>0$, then pick $\gamma_{n+1}$ s.t. $\gamma_{n+1}>\sup_{\alpha\in A_{n-1}} f_\alpha(n+1)$ and $B_{n+1}=\{\alpha\in B_n: f_\alpha(n+1)<\gamma_{n+1}\}$ has size $\mu^+$. Let $A_n$ be a set of $\mu_{n}$ many from $B_{n+1}$.

Then $\bigcup_n A_n$ has size $\kappa$ and is bounded by $n\mapsto \gamma_n$.

Learning project #1: a theorem

0th post in this series
1st post (updated with some more facts about IA structures at the end)
(edited 12/10/2014 to improve clarity)

We'll explain the proof of Theorem 27 of FMS:

Theorem 27: Suppose $\kappa$ is a supercompact cardinal and $\mu<\kappa$ is regular. Then in $V^{\rm{Col}(\mu,<\kappa)}$, the nonstationary ideal on $\mu$ is precipitous.

We'll follow the proof in FMS, trying to use Foreman's chapter for some new insights. I think the material is generally covered much better in the Handbook chapter than in the article.

Now we have Lemma 1 for proving ideals are precipitous, we can carry out this gameplan.

Let $G$ be a generic for $\rm{Col}(\mu,<\kappa)$, and let's think in the generic extension $V[G]$. We will show:

 Main Claim: Let $\mathfrak{A}$ be an expansion of $\langle H(\lambda),\in,\Delta\rangle$ (where $\lambda$ is some regular cardinal above, say, $2^{2^{2^\mu}}$). Then for almost all (in the sense of the nonstationary ideal) $N\prec\mathfrak{A}$ with $N\in \rm{IA}$, $|N|<\mu$, and $N\cap \mu$ an ordinal:

For all maximal antichains $\mathcal{A} \in N$ of $P(\mu)/NS$, there is an $A\in \mathcal{A}$ such that 

1. $N\cap \mu\in A$ and
2. letting $N^A$ equal the Skolem hull of $N\cup \{A\}$, $N^A\cap \mu=N\cap \mu$ in $\mathfrak{A}$.

Let's explain the usefulness of these conditions. To apply our Lemma 1, let $\mathcal{A}=\langle A \rangle$ be a tree of maximal antichains for $P(\mu)/\rm{NS}$ with underlying tree $T\subseteq \langle\mu^+\rangle^{<\omega}$. We will construct a branch $f:\omega\rightarrow \mu^+$ so that $\bigcap_n A_{f\upharpoonright n}\neq \emptyset$.

We'll construct the branch $f$ in $\omega$ stages. 

Take $N_0$ containing $\mathcal{A}$ as an element. Let $s_0=\langle\alpha_0\rangle$ be so that $A_{s_0}$ witnesses (1) and (2) with antichain equal to the first level of $\mathcal{A}$. From now on, use the notation $s_i:=\langle \alpha_0,\ldots, \alpha_{i}\rangle$. 

At the $i+1$st stage, let $N_{i+1}:=(N_i)^{\alpha_i}$ (the Skolem hull of $N_i\cup\{\alpha_i\}$ in $\mathcal{L}$). Pick $\alpha_{i+1}$ so that (1) and (2) are satisfied for $N=N_{i+1}$ and $s=\langle \alpha_0,\ldots, \alpha_{i+1}\rangle$ in the maximal antichain consisting of $\langle A_s:s \textrm{ extends }s_i\rangle$ together with some set not containing the single point $N\cap \mu$ (in order to do this construction, we will need to show that the property that $N_i\in\rm{IA}$ is maintained, which is Lemma 2 from last time). 

We have $N_i\cap \mu\in A_{s_i}$ for all $i$, but $N_i\cap\mu$ was always just equal to $N\cap \mu$, so $N\cap \mu$ would be in the intersection of all of the $A_{s_i}$. The key point of absorbing the index of the member of the antichain was so that the construction could continue in a coherent way, ensuring we are actually building a branch of $T$. (Foreman's chapter further develops these ideas, leading to the notion of "catching antichains".)

Let us now prove the main claim. Suppose there is a stationary $S\subseteq \rm{IA}$ of structures for which the conclusion of the main claim fails. The idea of the proof is that the supercompact gives us enough reflection of $S$ to catch the antichain. Let $\mathbb{P}=\rm{Col}(\mu,<\kappa)$ and let $j:V\rightarrow M$ be an $|H(\lambda)|$-supercompact embedding. Since $\mathbb{P}$ is $\kappa$-c.c., we can force to find $H\subseteq j(\mathbb{P})$ which contains $G$ as a subset, so we can extend $j$ to $\hat{j}:V[G]\rightarrow M[H]$ in $V[H]$.  

Subclaim: $S$ reflects to a set of size $\mu$, i.e., there is $Y\subseteq \lambda$ so that $\mu\subseteq Y$ and $|Y|=\mu$ so that $S\cap P(Y)$ is stationary in $P(Y)$.

Proof of Subclaim: By Lemma 3 from last time, $\hat{j}(S)\cap P(\hat{j}``H(\lambda))$ is stationary in $V[H]$. Now because of the collapse, $|\hat{j}``H(\lambda)|=\mu$ and $\mu\subseteq \hat{j}``H(\lambda)$ since $\kappa$ is the critical point. By closure of $M$ (and, it can be seen, of $M[H]$, $\hat{j}``H(\lambda)\in M[H]$. So ($\hat{j}$ of) the statement of the subclaim holds in $M[H]$, so by elementarity the subclaim holds in $V[G]$.

Returning to the proof of the main claim, let $Y$ be given by the subclaim and let $f:\mu\rightarrow Y$ be a bijection. Let $T:=\{\delta<\mu: f``\delta\in S\textrm{ and }\delta=f``\delta\cap\mu\}$. Then it is easy to see that $T$ is stationary, using the fact that $S$ reflects to $Y$. By maximality, there is $A\in\mathcal{A}$ such that $T\cap A$ is stationary. So we can find an IA structure $N'\prec \mathfrak{A}$ containing $A, f$ and with $N'\cap \mu \in T\cap A$. Take $N$ to be $f``(N' \cap\mu)$, so $N\in S$ by definition of $T$. By the second property in the definition of $T$, we also have $N\cap \mu=N'\cap \mu$. But $N'$ contains $A$, so the Skolem Hull of $N\cup\{A\}$ is contained in $N'$ and therefore has the same supremum below $\mu$, and would show that $N$ cannot be in $S$.

This finishes the proof. 

Learning project #1, part 1: preliminaries

0th post in this series (updated with some more motivation):
(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

1. $A_\emptyset = S$,
2. 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 $\mathcal{A}_n$, $n<\omega$. 

Lemma 1: (FMS p. 31, originally Jech--Prikry) $I$ is precipitous iff for every $S\in I^+$ and every tree $\mathcal{A}$ of maximal antichains for $P(Z)/I$ restricted to $S$, there is a branch $f$ of the tree so that $\cap_{n<\omega} A_{f\upharpoonright n}$ is nonempty.

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

1. $[S]\Vdash \dot{F}_n:Z\rightarrow V$,
2. $[S]\Vdash \dot{F}_n\in V$,
3. $[S]\Vdash ``M \vDash [\dot{F}_{n+1}]\in [\dot{F}_n]''$. 

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 $z\in A_t$.

If we have a tree of maximal antichains with just the first property, we can throw out sets in $I$ from each node to get the second property.

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

1. 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 $F(x)=y$.
2. 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.