UCI Summer School part 2: Duality Theorem (Monroe Eskew)

The Duality Theorem gives a general technique for forcing to make an ideal whose quotient algebra has various properties. It appears in Matthew Foreman's "Calculating quotient algebras of generic embeddings." My version of these notes omits a lot of the dots which indicate that certain objects are just names in a forcing extension. This is for aesthetic reasons, and hopefully does not lead to confusion.

Duality Theorem: Suppose $I$ is a precipitous ideal on $Z$ and $\mathbb{P}$ is any partial order. If: there is a further generic extension of the extension by $P(Z)/I$ so that if $j:V\rightarrow M\subseteq M\subseteq V[G]$ is the ultrapower embedding from $G\subseteq P(Z)/I$, there is $H\subseteq \mathbb{P}$ generic over $V$ and $\hat{H}\subseteq j(\mathbb{P})$ generic over $M$ and some extension of $j$ to $\hat{j}:V[H]\rightarrow M[\hat{H}]$.

Then: there is a $\mathbb{P}$-name for an ideal $J$ on $Z$ and a $P(Z)/I$-name for an ideal $K$ on $j(\mathbb{P})$ and a canonical isomorphism $$\iota:\mathcal{B}(\dot{\mathbb{P}\ast P(Z)/J})\equiv \mathcal{B}(P(Z)/I\ast j(\mathbb{P})/\dot{K}).$$


So a very general statement of lifting a generic ultrapower map to a forcing extension gives a useful isomorphism for computing $P(Z)/J$ in the generic extension by $\mathbb{P}$. We remark that in some cases, this will be an equivalence.

Note: In what follows, we tacitly identify all of the posets involved with their Boolean completions. Occasionally for emphasis, this identification will be explicit.

Proof: Assume (1). There is some $A\in I^+$ and some $P(Z)/I$-name for a forcing $\dot{\mathbb{Q}}$ so that
$$A\Vdash_{P(Z)/I}(\Vdash_\dot{\mathbb{Q}} \dot{H_0}\subseteq j(\mathbb{P}) \textrm{ is generic over }M \textrm{ and }H:=j^{-1}[H_0]\subseteq \mathbb{P}\textrm{ is generic over }V).$$

Note that the set $\{p\in\mathbb{P}:\quad \Vdash_{P(A)/I\ast \mathbb{Q}} j(p)\not\in H_0\}$ cannot be dense, since it is the complement of the generic $j^{-1}[H_0]$. So there is $p_0\in \mathbb{P}$ so that for all $p\le p_0$, $\| j(p)\in H_0 \|_{P(A)/I\ast \mathbb{Q}}\neq 0$. We will constrain ourselves to work below this $p_0$ in $\mathbb{P}$ and below $A$ in $P(Z)/I$. For simplicity, assume that $A=Z$ and $p_0=1_\mathbb{P}$.

In $V^{P(Z)/I}$, define
$$K=\{p\in j(\mathbb{P}):\quad \Vdash_{\mathbb{\dot{Q}^G}} p\not\in H_0\}.$$
Let $G\ast h$ be generic for $P(Z)/I\ast j(\mathbb{P})/K$. From the $j(\mathbb{P})/K$-generic $h$, we can define a $j(\mathbb{P})$-generic $\hat{H}=\{p:[p]_K\in h\}$.

Claim: The following properties of $H_0$ are also true of $\hat{H}$:

1. $\Vdash_{P(Z)/I\ast \mathbb{Q}} \hat{H}$ is $j(\mathbb{P})$-generic over $M$.
2. $\Vdash_{P(Z)/I\ast \mathbb{Q}} j^{-1}[\hat{H}]$ is $\mathbb{P}$-generic over $V$.
3. For all $p\in \mathbb{P}$, $\not\Vdash_{P(Z)/I\ast \mathbb{Q}} j(p)\not\in \hat{H}$.

Proof of Claim: For (1), suppose $D\in M$ is open dense in $j(\mathbb{P})$. Then $\{[d]_K:d\in D\textrm{ and }d\not\in K\}$ is dense in $j(\mathbb{P})/K$. Otherwise, there would exist $p\in j(\mathbb{P})/K$ so that $p\wedge d\in K$ for all $d\in D$. But this is impossible because we could then force with $\mathbb{Q}$ over $V[G]$ to get a generic $H_0$ containing $p$ (as $p\not\in K$), and then $H_0\cap D=\emptyset$, contradicting genericity of $H_0$ over $M$.

The remaining parts of the claim can be checked similarly, and are left as an exercise. $\Box$.

Now let $e:\mathbb{P}\rightarrow \mathcal{B}(P(Z)/I\ast \dot{j(\mathbb{P})/K})$ be defined by $e(p)=\|j(p)\in \hat{H}\|$.

By (3) of the claim above, $\mathrm{ker}(e)=0$. Also $e$ preserves Boolean operations simply by the elementarity of $j$. By (2) of the claim, $e$ is a regular embedding (maps maximal antichains pointwise to maximal antichains). 

Exercise: $e:\mathbb{P}\rightarrow \mathbb{Q}$ is a regular embedding iff for every $q\in\mathbb{Q}$ there is $p\in \mathbb{P}$ so that for every $p'\le p$, $e(p')$ is compatible with $q$.

Thus, if $H\subseteq \mathbb{P}$ is generic over $V$, we can force with the quotient $\mathcal{B}(P(Z)/I\ast \dot{j(\mathbb{P})/K})/e"H$ over $V[H]$ to obtain a generic $G\ast h$ for $P(Z)/I\ast j(\mathbb{P})/K$. By the definition of $e$, we have $j_G"H\subseteq \hat{H}$, where $\hat{H}$ is defined from $h$ as before. So we can extend the embedding $j_G$ to $\hat{j}:V[H]\rightarrow M[\hat{H}]$.

In $V[H]$ we can finally define $J=\{A\subseteq Z:1\Vdash [\mathrm{id}]\not\in \hat{j}(A)\}$, where the forcing is with the quotient $\mathcal{B}(P(Z)/I\ast \dot{j(\mathbb{P})/K})/e"H$. In $V$, let

$$\iota(p,\dot{A})=e(p)\wedge \|[id]\in\hat{j}(\dot{A})\|.$$

Exercise: $\iota$ is order and incompatibility preserving.

It remains to show that the range of $\iota$ is dense. So take an arbitrary $(B,\dot{q})\in P(Z)/I\ast j(P)/K$. By strengthening this condition, we may assume without loss of generality that there is $f:Z\rightarrow \mathbb{P}$ in $V$ so that $B\Vdash [[f]_M]_K=\dot{q}$.

By regularity of $e$ (using the characterization in the exercise), there is a $p$ so that for all $p'\le p$, $e(p')\wedge (B,\dot{q})\neq 0$. Let $\dot{A}$ be a $\mathbb{P}$-name for a subset of $Z$ such that $p \Vdash \dot{A}=\{z\in B: f(z)\in H\}$ and $\neg p \Vdash \dot{A}\in J^+$.

We check that $(p,\dot{A})$ is actually a condition, which involves checking that $p\Vdash \dot{A}\in J^+$. So take a generic $G\ast \dot{h}$ containing $e(p)\wedge (B,\dot{q})$ (which is nonzero by choice of $p$). Then clearly $B\in G$ and since $[[f]_M]_K=q\in h$, we have $[f]_M=j(f)([\mathrm{id}])\in\hat{H}$. Therefore $[\mathrm{id}]\in \hat{j}(A)$, so this generic $G\ast \dot{h}$ shows that it is not forced by $\mathcal{B}(P(Z)/I\ast \dot{j(\mathbb{P})/K}/e"H$ that $[\mathrm{id}]\not\in \hat{j}(A)\}$.

By definition $\iota:=\iota(p,\dot{A})$ forces $j(p)\in \hat{H}$ and $[\mathrm{id}]\in\hat{j}(\dot{A})$. Since $\hat{j}$ extends $j$ and it's forced that $\dot{A}\subseteq B$, $\iota$ must force $B\in G$. And since $j(p)\Vdash_{j(\mathbb{P})} j(\dot{A})=j(\{z:j(f)(z)\in \hat{H}\})$, $\iota$ must force $\dot{q}=[j(f)(\mathrm{id})]_K\in h$. $\Box$

Remark: Suppose $K$ as in the Duality Theorem is forced to be principal, i.e., there is $m$ so that $$\Vdash K=\{p\in j(\mathbb{P}:p\le \neg m\}.$$

Then the Duality Theorem is easily seen to be an equivalence.

We can compute some nice properties of the ideal $J$ arising from the previous theorem.

Proposition: Using the notation of the previous theorem, $J$ is forced to be precipitous, with the same completeness as $I$. If $I$ is normal, then $J$ is also normal. Also, if $\bar{G}\subseteq P(Z)/J$ is generic over $V[H]$ and $G\ast h=\iota[H\ast \bar{G}]$ and $\hat{j}:V[H]\rightarrow M[\hat{H}]$ are as before, then $V[H]^Z/{\bar{G}}=M[\hat{H}]$ and $\hat{j}$ is the ultrapower embedding.

Finally, we relate $J$ to the ideal in $V[H]$ generated by $I$.

Proposition: Suppose $K$ as in the Duality Theorem is forced to be principal, with $m$ so that $\Vdash K=\{p\in j(\mathbb{P}):p\le \neg m\}.$ Suppose further that there exist $f$ and $A$ so that $A\Vdash \dot{m}=[f]_G$ and $\dot{B}$ is a $\mathbb{P}$-name for $\{z\in A: f(z)\in \dot{H}\}$. Then $\bar{I}\upharpoonright B=J\upharpoonright B$, and $A\setminus B\in J$, where $\bar{I}$ is the ideal in $V[H]$ generated by $I$.

UCI Summer School part 1: Basics (Monroe Eskew)

The UCI summer school in set theory just finished, and I'll be posting my notes on this blog. They will not be added in chronological order, and the final product will be very different from the actual presentation, assume that all errors were added by me.

There are some exercises which we solved during problem sessions during the day; I will preserve these in the text and can post solutions as well if there is interest.

We start off with Monroe Eskew's lectures on the duality theorem. 

Basic facts: This section is meant to be supplemented by my earlier postings about ideal properties. The basic situation is that $I$ is an ideal on $Z\subseteq P(X)$, $G$ is a generic for the forcing $P(Z)/I$, and $j:V\rightarrow M=V^Z/G$ is the generic ultrapower by $G$.

Definition: An ideal $I$ has the disjointing property if every antichain $A$ in $P(Z)/I$ has a pairwise disjoint system of representatives.

Exercise: If $I$ is a $\kappa$-complete, $\kappa^+$-saturated ideal, then $I$ has the disjointing property.  If $I$ is a normal ideal on $Z\subseteq P(X)$, and $I$ is $|X|^+$-saturated, then $I$ has the disjointing property. 

Lemma: Suppose $A\in I^+$ and $I$ has the disjointing property, and $[A]\Vdash \dot{\tau} \in V^Z/\dot{G}$. Then there is $f:Z\rightarrow V$ such that $A\Vdash \tau = [f]_G$.

Proof: Exercise.

Lemma: Suppose $I$ is a countably closed ideal with the disjointing property. Let $\kappa=\mathrm{crit}(j)$. Then $I$ is precipitous, and the generic ultrapower $M$ is closed under $\kappa$-sequences from $V[G]$.

Proof: Precipitousness follows from the combinatorial characterization of precipitousness. Suppose that $\Vdash \langle \tau_\alpha:\alpha<\kappa\rangle \subseteq V^Z/G$.  By the previous lemma, for all $\alpha<\kappa$, there is $f_\alpha:Z\rightarrow V$ in $V$ such that $\Vdash \tau_\alpha=[f_\alpha]_G$ and $k:Z\rightarrow \mathrm{ON}$ in $V$ such that $\Vdash \kappa=[k]_G$. Finally, take $f:Z\rightarrow V$ given by  $f(z)=\langle f_\alpha(z):\alpha<k(z)\rangle$. In $M$, $[f]_G=\langle \tau_\alpha:\alpha<\kappa\rangle$. $\Box$

Definition: $I$ is $(\lambda,\kappa)$-presaturated if for any sequence of antichains $\langle A_\alpha:\alpha<\gamma<\lambda\rangle$, the set $\{X:\forall\alpha<\gamma\, |\{a\in A_\alpha:a\cap x\in I^+\}|<\kappa\}$ is dense.

Exercise: $(\lambda^+, \kappa^+)$-presaturation and $\kappa$-completeness imply that the generic ultrapower $M$ is closed under $\lambda$-sequences from $V[G]$. Also show this for normal and $(\lambda^+,|X|^+)$-presaturated ideals.

We will use the notation $\mathcal{B}(\mathbb{P})$ to denote Boolean completion of a poset. As we have seen in previous posts, this is very handy when dealing with generic ultrapowers.

Exercise: $\mathbb{P}$ and $\mathbb{Q}$ are separative posets. The following are equivalent: 

1. $\mathcal{B}(\mathbb{P})=\mathcal{B}(\mathbb{Q})$.
2. There is a $\mathbb{P}$-name $\dot{h}$ and a $\mathbb{Q}$-name $\dot{g}$ so that $$\Vdash_\mathbb{P} \dot{h} \textrm{ is }\mathbb{Q}-\textrm{generic over }V,$$ $$\Vdash_\mathbb{Q} \dot{g} \textrm{ is }\mathbb{P}-\textrm{generic over }V,$$ and$$\Vdash_\mathbb{P} \dot{g}^{\dot{h}^\dot{G}}=\dot{G} \textrm{  and  } \Vdash_\mathbb{Q} \dot{h}^{\dot{g}^\dot{H}}=\dot{H}.$$

Tight stationarity and tree-like scales

My article, "Tight stationarity and tree-like scales," is now available online in the Annals of Pure and Applied Logic.

You can also find the final version here.

I have some slides on this topic from last year's BEST conference available here. (Coincidentally, I'm now at the 2015 BEST conference :)

Spencinar: Morley's categoricity theorem, part 1

Unfortunately, I missed posting about some interesting lectures given at the Very Informal Gathering, and the Southern California logic meeting at Caltech. We'll return with this series of posts following Spencer Unger's talks at the first two seminars of the quarter. There may be a larger amount of errors than usual in my transcription because of my limited knowledge of model theory. Basic model theory will be helpful to follow these notes, but I'll do my best to fill in the definitions. The notation follows Marker's text closely.

A general question is:
Given a first order theory $T$, what can we say about the number of non-isomorphic models of $T$ of size $\lambda$, as $\lambda$ varies through the cardinals?

This is addressed by Shelah's classification theory. Historically, the first step was Morley's categoricity theorem (1965) which states:

• If $T$ is a first-order theory in a countable language and $T$ is categorical in some uncountable cardinal, then it is categorical in all uncountable cardinals.

The proof of this theorem, which was the content of the lectures, involves the ideas of stability, indiscernibles, saturated models, and splitting types. It's somewhat different from the one I recall from the basic model theory class I took many years ago.

To begin with, we can define a rank $R$ on sets of formulas (i.e., types) with parameters in some large saturated model $\mathcal{C}$ (the "monster model"). Let $p$ be a set of formulas. Then, working in $\mathcal{C}$,

• $R(p)=-1$ if $p$ is not satisfiable.
• $R(p)\ge 0$ if $p$ is satisfiable.
• $R(p)\ge \alpha+1$ if for any finite $p_0\subseteq p$, there is a formula $\psi$ such that $R(p_0\cup\{\psi\})\ge \alpha$ and $R(p\cup\{\neg \psi\})\ge \alpha$.

The rank $R(p)$ is then defined inductively to be the minimum value so that all of the above conditions are satisfied. Note: this isn't what is known as Morley rank. Andrew Marks noticed that this rank is the Cantor-Bendixson rank on the Stone space of complete types.

Some properties of the rank:

• (Monotonicity) If $p\vdash q$ then $R(p)\le R(q)$.
• (Finite character) For every $p$, there is a finite $p_0\subseteq p$ so that $R(p)=R(p_0)$. 
• Proof: If $R(p)=\alpha$, there is a finite $p_0$ witnessing $R(p)\not\ge \alpha+1$, and monotonicity gives the reverse inequality.
• (Invariance) If $f$ is an automorphism of $\mathcal{C}$, then $R(p)=R(f(p))$.
• Proof is by induction on the rank.

Definition: $T$ is $\lambda$-stable if for any $A\subseteq C$ of size $\lambda$, $|S^n(A)|\le \lambda$. Here $S^n(A)$ is the set of complete $n$-types over $A$, that is, maximal satisfiable sets of formulas with parameters over $A$ and free variables $x_1,\ldots, x_n$.

Theorem: Let $T$ be a theory in a countable language. Then $T$ is $\aleph_0$-stable iff $R(\{\bar{x}=\bar{x}\})<\infty$, i.e., has some ordinal value. In fact, if these equivalent conditions hold, then $R(\{\bar{x}=\bar{x}\})<\omega_1$.

Proof: Suppose $R(\{\bar{x}=\bar{x}\})\ge \omega_1$. We will inductively construct a complete binary tree whose nodes are finite sets of formulas $p_\eta$ for $\eta\in 2^{<\omega}$ and whose branches will give $2^{\aleph_0}$ many different complete types, contradicting stability. For the construction, we will maintain that $R(p_\eta)\ge\omega_1$.

Let $p_\emptyset=\{\bar{x}=\bar{x}\}$. Given $p_\eta$, there are formulas $\psi_{\eta,i}(\bar{x},\bar{a}_{\eta,i})$ for each $i<\omega_1$ witnessing that $R(p_\eta)\ge i+1$. Using the fact that the language of $T$  is countable together with $\aleph_0$-stability, there are $\psi$ and $t$ so that for an unbounded set of $i<\omega_1$, $\psi=\psi_{\eta,i}$ and $t=\mathrm{tp}(\bar{a}_{\eta,i})$ (the set of all formulas satisfied by $\bar{a}_{\eta,i}$, over the finitely many parameters in $p_\eta$). Pick $a_\eta$ realizing $t$. Then define 

• $p_{\eta\circ 0}=p_\eta\cup\{\psi(\bar{x},\bar{a}_\eta)\}$
• $p_{\eta\circ 1}=p_\eta\cup\{\neg\psi(\bar{x},\bar{a}_\eta)\}$

Using invariance of rank under automorphisms and the fact that in $\mathcal{C}$ for any two tuples realizing the same type there exists an automorphism sending one to the other, the ranks of these sets of formulas are at least $i$ for an unbounded set of $i<\omega_1$, so they must be $\ge\omega_1$, allowing the construction to continue. The branches of the tree we construct can be extended to complete types, and the construction ensures that each of these complete types is distinct (for two different branches, look at the first place of disagreement).

For the converse, assume $R(\bar{x}=\bar{x})<\infty$ and $T$ is not stable. Then there are $A\subseteq \mathcal{C}$, $|A|\le \lambda$ such that there are distinct $\{p_i:i<\lambda^+\}\subseteq S^n(A)$. By assumption, for any $i<\lambda^+$, there is $\alpha_i$ so that $R(p_i)=\alpha_i$. By finite character, there is $q_i\subseteq p_i$ finite with $R(q_i)=R(p_i)=\alpha_i$. By counting, $|\{q_i:i<\lambda^+\}|\le \omega$. Let $i\neq j$ be such that $q_i=q_j$, so $p_i,p_j$ are different completions of $q_i=q_j$. This means there is a formula $\psi$ with $\neg\psi\in p_i$ and $\psi\in p_j$. This gives $R(q_j)\ge R(p_j)+1$, a contradiction to the choice of $q_j$. $\Box$.

Spencinar: Forcing clubs in stationary subsets of $P_\kappa(\lambda)$

The first part of this talk follows closely the paper "Forcing closed unbounded sets" by Uri Abraham and Saharon Shelah ([AS]). The second part will follow the paper "Nonsplitting subset of $P_\kappa(\kappa^+)$ by Moti Gitik ([G]).

We have previously seen that given a stationary $S\subseteq \omega_1$, there is a forcing which has the $\omega_2$-c.c. and is $<\omega_1$-distributive (and hence preserves all cardinals) which adds a club $C\subseteq S$, namely the forcing of closed bounded subsets of $S$, ordered by end-extension (Baumgartner--Harrington--Kleinberg).

This generalizes to higher regular cardinals $\kappa$ assuming some cardinal arithmetic (GCH with $\kappa$ the successor of a regular cardinal suffices) and fatness of $S$, which says that for any club $E\subseteq \kappa$, $S\cap E$ contains closed subsets of arbitrarily large order type $<\kappa$ (this is a result of J. Stavi).

The focus of this talk will be to obtain analogues for these theorems for subsets of $P_\kappa(\lambda)=\{x\subseteq \lambda: |x|<\kappa\}$.

This situation is not as clear as for subsets of ordinals, and in fact there are some sets whose stationarity is quite absolute (By stationary and club I mean here in the sense of Jech, see this previous post). For example, Theorem 6 in [AS] gives a stationary subset of $P_{\aleph_1}(\omega_2)$ whose stationarity is preserved by any forcing which preserves $\aleph_2$:

Theorem: Let $W\subseteq V$ be a transitive inner model such that $\omega_2^W=\omega_2^V$, and let $S=(P_{\aleph_1}(\omega_2))^W$. Then $S$ is stationary in $V$.

Proof: The idea of the proof is to fix an arbitrary club $C\subset P_{\aleph_1}(\omega_2)$ and find submodels which are ordinals, or somehow coded by ordinals in $W$. Then these submodels will automatically be in $S$.

By Kueker's theorem, there is a function $F:[\omega_2]^{<\omega}\rightarrow \omega_2$ so that any $x\in P_{\aleph_1}(\omega_2)$ closed under $F$ is in $C$ (this uses the fact that we're working on $\aleph_1$ and $\aleph_2$, but this is a minor technical point). Let $\alpha$ be an ordinal closed under $F$. If $\alpha$ is countable, then we're done; otherwise $\alpha$ has cardinality $\aleph_1$, hence also $W$-cardinality $\aleph_1$, so fix a bijection $h:\omega_1\rightarrow \alpha$ in $W$. Then there is $\xi<\omega_1$ with $h[\xi]$ closed under $F$, and $h[\xi]$ is in $W$ and hence in $S$, so we're done again. $\Box$

In [G], it's shown that any forcing which adds reals to $W$ destroys the club-ness of $S$.

This paper gives some examples of sets whose stationary can be destroyed. We won't go through those results here. Instead, we now turn towards a result in [G] which also gives examples of this kind.

Let $\kappa$ be supercompact in $W$, and let $V$ be obtained from $W$ by Radin forcing. We won't need to know much about Radin forcing, just:

• the continuum function and all cardinalities are preserved,
• there is a club $C\subseteq \kappa$ of $W$-inaccessible cardinals,
• $\kappa$ remains (sufficiently) supercompact in $V$. 

From now on, work in $V$. Let $A$ be any subset of the set of all $t\in P_\kappa(\kappa^+)\cap W$ such that $V\vDash ``|t| \textrm{ is a successor cardinal}"$.

We will find a forcing that adds a club in $S:=P_\kappa(\kappa^+)\setminus A$ which is $<\kappa$-distributive (this is crucial, so that $P_\kappa(\kappa^+)$ itself does not change), and has the $\kappa^+$-c.c.

Define the poset $\mathbb{P}$ to be the collection of subsets of $S$ of size $<\kappa$ which have a maximum element and are closed under increasing unions, ordered by end-extension. 

Claim: $\mathbb{P}$ is $<\kappa$-distributive.

Let $\langle D_\beta: \beta<\alpha\rangle$ be a sequence of dense open subsets of $\mathbb{P}$ for some $\alpha<\kappa$.  Then we will prove that $\bigcap_{\beta<\alpha} D_\beta$ is also dense. So let $x\in \mathbb{P}$.

Let $M\prec (H(\theta),\in,\triangleleft,A,\langle D_\beta\rangle,x,\ldots)$ be an elementary substructure of size $\kappa$ so that $M\cap \kappa^+\in \kappa^+$ and $M$ is closed under $<\kappa$-sequences. Let $g\in W$ be a bijection from $\kappa$ onto $M\cap \kappa^+$. Using $<\kappa$-closure of $M$, construct $\langle M_i:i<\kappa\rangle$ a continuous IA chain of elementary submodels of $M$ of size $<\kappa$ so that $\alpha+1\subseteq M_0$ and $g``i\subseteq M_i$. There is a club $E\subseteq \kappa$ such that for every $i\in E$, $i\in C$ and $M_i\cap \kappa^+=g``i$. Let $\langle i_\beta:\beta<\kappa\rangle$ be the increasing enumeration of $E$, and let $N_\beta=M_{i_\beta}$.

We will inductively construct a decreasing sequence of conditions $\langle x_\beta:\beta\le \alpha\rangle$ such that:

•  $x_0\le x$,
•  $x_{\beta+1}\in D_\beta$,
•  $N_\beta\cap \kappa^+=\max(x_\beta)$,
•  $x_\beta\in N_{\beta+1}$.

Take $x'_0\in N_0$, $x'_0\le x$, and let $x_0=x'_0\cup\{N_0\cap \kappa^+\}$.

Now we construct $x_\beta$, assuming that $x_\gamma$ has been constructed for every $\gamma<\beta$. If $\beta=\gamma+1$ for some $\gamma$, then pick $x'_\beta$ to be the $\vartriangleleft$-least in $N_\beta\cap D_\beta$ extending $x_\gamma$, and define $x_\beta=x'_\beta\cup\{N_\beta\cap \kappa^+\}$. This is a valid condition since $N_\beta\cap \kappa^+$ has $W$-cardinality in $C$, so it can't be a member of $A$.

If $\beta$ is limit, then let $x'_\beta$ be the closure of $\bigcup \{x_{\gamma}:\gamma<\beta\}$ under increasing unions, and $x_\beta=x'_\beta\cup\{N_\beta\cap \kappa^+\}$. By internal approachability, $x_\beta\in N_{\beta+1}$.

It is easy to see that:

Claim: For every $\gamma<\beta$, $x_\beta$ is an end-extension of $x_{\gamma}$.

We now check that $x_\beta\in \mathbb{P}$. If not, then there is $t\in x_\beta\cap A$. Since $t\subseteq N_\beta\cap \kappa^+$, we must have $|t|^W\le i_\beta$. Furthermore, $|t|^W$ is a successor cardinal of $W$ and $i_\beta$ is W-inaccessible, so $|t|^W<i_\beta$.

Since $i_\beta$ is $W$-regular, there is some $\gamma<\beta$ with $t\subseteq g``i_\gamma=N_\gamma\cap \kappa^+$. But this is impossible by the claim.  

The $\kappa^+$-c.c. follows by a standard $\Delta$-system argument.

Gitik used this forcing as the building block of an iteration to produce a stationary subset $Z$ of $P_\kappa(\kappa^+)$ so that the non-stationary ideal restricted to that stationary set is $\kappa^+$-saturated, which is very interesting in light of results that show that the whole nonstationary ideal is not saturated. Another way to look at this is that it shows the consistency of the failure of a natural analogue of Solovay's splitting theorem for stationary subsets of $P_\kappa(\lambda)$.

Assuming that $\kappa$ is supercompact, the set $Z$ of all $t\in P_\kappa(\kappa^+)\cap W$ such that $W\vDash ``|t| \textrm{ is a successor cardinal}"$ is stationary. The idea of the iteration is that we will destroy the stationary of the "bad sets" to while maintaining the stationary of $Z$. Maintaining the stationarity of $Z$ is achieved through extending the supercompactness embedding to the final model, but this is not an easy task in this case.

An earlier version of this post had several occurrences of $V$ which should have been $W$.

Good and bad terminology in mathematics

Happy holidays! I hope everyone is having a restful vacation. Here's a post which is just for a bit of fun---don't take it too seriously!

A lot of people complain about bad terminology in mathematics. I don't really consider myself to be one of those people, but I have to admit they have a point. The choice of a word we use to represent a concept greatly influences the way we read and perceive it. Here are examples of terminology I like and dislike, with some whimsical alternatives for fun. It turned out to be much harder than I expected to come up with good terminology. Leave a comment if you can think of something to add to these lists.

Like

• Adjoint: This word seems to only exist in mathematics. It captures a general pattern that appears frequently in different contexts; several formally related concepts share this terminology. Appropriately, it suggests pairing or duality.
• Club set: Short for "closed and unbounded". This neat portmanteau helps emphasize that the conjunction of the two properties is somehow more the sum of its parts.
• Compactness: This is such a carefully nuanced word for a subtle and fundamental property which occurs in many different contexts in mathematics, although I think it can take some getting used to. It really captures the idea of some kind of finitary character, or a strong notion of "smallness".
• Mouse, weasel: Inner model theory words that give a mischievous flavor to a difficult subject.
• Mixing: from ergodic theory. It's a gerund, which helps describe this dynamical situation well.
• Rational numbers: This is a nice play on words. It makes $\mathbb{Q}$ feel very concrete and friendly and is probably derived from the representation of these numbers as ratios of integers. 
• Supercompact cardinal: sometimes terminology doesn't totally make sense, it just sounds cool. I have to admit that hearing this word early made me curious about set theory.

Dislike

• Amenable group: This seems to be a pun that only makes sense with British pronunciations. According to Wikipedia, the original name given by von Neumann was "messbar", or "measurable" in English. Of course, this name is even worse. Alternatives: Maybe something like "well-measured" would be better.
• Antichain: It's actually a catchy word, if only it didn't have two closely related but nonequivalent meanings. In order theory people use it to mean a collection of pairwise incomparable elements of a poset, while set theorists would say it is a collection of pairwise incompatible elements. Let the order theory people take back the outdated term "Sperner system".
• Cardinal collapsing: This term from set theory is really convenient and probably no one else in the world has a problem with it. My gripe is that it's somewhat inaccurate and easily replaced by simpler terms. "Collapsing" a cardinal refers to a forcing extension where a particular ordinal is no longer a cardinal, in other words, its cardinality is decreased (as measured by ordinals) in the extension. The word "collapse" is a bit overloaded (it seems better suited to describe things like Mostowski collapse). Alternatives: "decardinalizing", "cardinality decreasing".
• Commutative diagram: To commute just means to go from one place to another. This got twisted into the property that $ab=ba$; I guess I can kind of see that they are moving past each other. So we can also use this for functions $f,g:X\rightarrow X$ to say $fg=gf$ (a special kind of commutative square). But what about commutative triangles, some functions that satisfy $f=gh$? Alternatives: "freely composing", or maybe something made-up and descriptive like "ambicompositional".
• Countable chain condition (c.c.c.): This one from set theory is unpopular among some people I know, since it's a property of posets that means there are no uncountable antichains (and I don't mean Sperner systems here). I saw one paper (Abraham--Shelah: Forcing closed unbounded sets) which called this the c.a.c., for countable antichain condition, which seems like a reasonable alternative. I recall that there's actually a good historical reason for this terminology, though I'm not sure exactly what that was.
• Maths: OK, so it's not really in the spirit of the other things, and maybe its just my American crudeness, but this sounds totally wrong to me.
• One-to-one: I thought it was OK until someone pointed out to me that it sounds like each member of the domain maps to a unique member of the codomain (what some people call single-valued). This has been the source of some confusion in classes I've TA'ed. Alternatives: just use the already existing technical-sounding words "injective" and "surjective."
• Proper forcing: It's just a bit too vague and authoritarian for me. Alternatives: Maybe something like "internally generic forcing".

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 $n\mapsto \gamma_n$.