ON AUTOMORPHISMS OF

JAKOB KELLNER; SAHARON SHELAH; ANDA RAMONA TĂNASIE

doi:10.1017/jsl.2024.37

ON AUTOMORPHISMS OF ${\mathcal {P}}(\lambda )/[\lambda ]^{<\lambda }$

Part of: Set theory

Published online by Cambridge University Press: 16 May 2024

and

JAKOB KELLNER*: Affiliation:
INSTITUTE OF DISCRETE MATHEMATICS AND GEOMETRY TECHNISCHE UNIVERSITÄT WIEN (TU WIEN) VIENNA AUSTRIA URL: http://dmg.tuwien.ac.at/kellner/
SAHARON SHELAH: Affiliation:
EINSTEIN INSTITUTE OF MATHEMATICS THE HEBREW UNIVERSITY OF JERUSALEM JERUSALEM ISRAELand RUTGERS UNIVERSITY DEPARTMENT OF MATHEMATICS NEW JERSEY USA E-mail: [email protected] URL: http://shelah.logic.at/
ANDA RAMONA TĂNASIE: Affiliation:
INSTITUT FÜR INFORMATIK FH WIENER NEUSTADT WIENER NEUSTADT AUSTRIA E-mail: [email protected]
*: E-mail: [email protected]

Article contents

Abstract
Introduction
Definitions
Under MA, every automorphism is trivial for $\omega _1\le \lambda
For measurables, GCH implies a nontrivial automorphism
For inaccessible $\lambda $, all automorphisms can be densely trivial
Funding
Footnotes
References

Rights & Permissions

Abstract

We investigate the statement “all automorphisms of ${\mathcal {P}}(\lambda )/[\lambda ]^{<\lambda }$ are trivial.” We show that MA implies the statement for regular uncountable $\lambda <2^{\aleph _0}$, that the statement is false for measurable $\lambda $ if $2^\lambda =\lambda ^+$, and that for “densely trivial” it can be forced (together with $2^\lambda =\lambda ^{++}$) for inaccessible $\lambda $.

Keywords

rigid Boolean algebras trivial automorphisms

MSC classification

Primary: 03E40: Other aspects of forcing and Boolean-valued models 03E35: Consistency and independence results

Type: Article
Information: The Journal of Symbolic Logic , First View , pp. 1 - 37

DOI: https://doi.org/10.1017/jsl.2024.37 [Opens in a new window]
Creative Commons: This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright: © The Author(s), 2024. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

1. Introduction

We investigate automorphisms of Boolean algebras of the form

$$\begin{align*}P^\lambda_\kappa:={\mathcal{P}}(\lambda)/[\lambda]^{<\kappa}.\end{align*}$$

The instance $P^{\omega }_{\omega }$ , i.e., ${\mathcal {P}}(\omega )/\mathrm {FIN}$ , has been studied extensively for many years.Footnote ¹ One can study variants for uncountable cardinals $\lambda $ . Unsurprisingly, the behaviour here tends to be quite different to the countable case. One moderately popularFootnote ² such generalisation is $P^\lambda _{\omega }$ . Here, we study another obvious generalization of the countable case, $P^\lambda _{\lambda }$ . Some results for general $P^\lambda _\kappa $ can be found in [Reference Larson and McKenney5].

The main result of the paper is:

(Theorem A, Theorem 5.2)

$$ \begin{align} \nonumber & \mathit{The\ following\ is\ equiconsistent\ with\ an\ inaccessible}{:}\ \lambda\ \mathit{is\ inaccessible}, 2^\lambda\ \mathit{is } \\& \nonumber \lambda^{++},\ \mathit{and\ all\ automorphisms\ of}\ P^\lambda_{\lambda}\ \mathit{are\ densely\ trivial.} \\ \end{align} $$

Here, $2^\lambda>\lambda ^{+}$ is necessary, at least for measurables:

(Theorem B, Theorem 4.1)

$$ \begin{align} \mathit{If}\ \lambda\ \mathit{is\ measurable\ and}\ 2^\lambda=\lambda^+,\ \mathit{then\ there\ is\ a\ nontrivial\ automorphism\ of}\ P^\lambda_{\lambda}. \end{align} $$

Remark 1.1. From [Reference Shelah and Steprāns12, Lemma 3.2] it would follow that Theorem B holds even when “measurable” is replaced by just “inaccessible.” However, the proof there turned out to be incorrect.Footnote ³

For $\lambda $ below the continuum we get the following result under Martin’s Axiom (MA). More explicitly, MA $_{=\lambda }(\sigma \text{-centered})$ is sufficient, which is the statement that for any $\sigma $ -centered poset P and ${\le }\lambda $ many open dense sets in P there is a filter G meeting all these open sets:

(Theorem C, Theorem 3.1)

$$ \begin{align} \nonumber & \mathit{For}\ \aleph_0<\kappa\le\lambda<2^{\aleph_0}\ \mathit{and}\ \kappa\ \mathit{regular},\ MA_{=\lambda}(\sigma\mathit{-centered})\ \mathit{implies\ that\ every}\\\nonumber & \mathit{automorphism\ of}\ P^\lambda_{\kappa}\ \mathit{is\ trivial.}\\ \end{align} $$

Larson and McKenney [Reference Larson and McKenney5] showed the same under MA $_{\aleph _1}$ for the case $\lambda =2^{\aleph _0}$ and $\kappa =\aleph _1$ .

Contrast this to the case $\lambda =\kappa =\omega $ : Due to results of Veličković, Steprāns, and the second author, “Every automorphisms of ${\mathcal {P}}(\omega )/[\omega ]^{<\omega }$ is trivial” is implied by PFA [Reference Shelah and Steprāns10], in fact even by MA+OCA [Reference Veličković15], but not by MA alone [Reference Veličković15] (not even for “somewhere trivial” [Reference Shelah and Steprāns11]).

We start by introducing some notation and basic results in Section 2 (page 2).

The following sections are independent of each other:

In Section 3 (page 3) we show Theorem C, which we state as Theorem 3.1; in Section 4 (page 6), we show Theorem B, i.e., Theorem 4.1; and finally in the main part, Section 5 (page 7) we develop some forcing notions to prove Theorem A, i.e., Theorem 5.2.

2. Definitions

We always assume that $\lambda $ is a cardinal and $\kappa \le \lambda $ is regular.

• The case $\kappa =\aleph _0$ or $\lambda =\aleph _0$ is included only for completeness sake in the following definitions.
• In Section 3 we will assume that $\aleph _1\le \kappa \le \lambda < 2^{\aleph _0}$ .
• In Section 4 we assume that $\lambda $ is measurable and $\kappa =\lambda $ .
• In Section 5 we assume that $\lambda $ is inaccessible and $\kappa =\lambda $ .

Notation:

• We investigate the Boolean algebra (BA) $P^\lambda _\kappa :={\mathcal {P}}(\lambda )/[\lambda ]^{{<}\kappa }$ , i.e., the power set of $\lambda $ factored by the ideal of sets of size ${<}\kappa $ .
• For $A\subseteq \lambda $ , we denote the equivalence class of A with $[A]$ . We set $\mathbb 0:=[\emptyset ]$ .
• $A\subseteq ^* B$ means $|B\setminus A|<\kappa $ , analogously for $A=^*B$ ; and “for almost all $\alpha \in A$ ” means for all but ${<}\kappa $ many in A. In particular, $A=^*\lambda $ means $A\subseteq \lambda $ and $|\lambda \setminus A|<\kappa $ .
• We denote the BA-operations in $P^\lambda _\kappa $ with $x \vee y$ , $x \wedge y $ and $x^c$ (for the complement). So we have $[A]\vee [B]=[A\cup B]$ , $[A]\wedge [B]=[A\cap B]$ , and $[A]^c=[\lambda \setminus A]$ .
• A function $\phi : P^\lambda _\kappa \to P^\lambda _\kappa $ is a BA-automorphism (which we will just call automorphism), if it is bijective, compatible with $\wedge $ and the complement, and satisfies $\phi (\mathbb 0)=\mathbb 0$ .
• Preimages of a function f are denoted by $f^{-1}x$ , images by $f"x$ .
• We sometimes identify $\eta \in 2^\lambda $ with $\eta ^{-1}\{1\}\subseteq \lambda $ without explicitly mentioning it, by referring to $\eta $ as element of $2^\lambda $ or of $P(\lambda )$ .

Let us note that $P^\lambda _\kappa $ is ${<}\kappa $ -completeFootnote ⁴ and $\lambda ^+$ -cc. Also, any automorphism $\phi $ is closed under ${<}\kappa $ unions: $\phi (\bigvee _{i\in I}[A_i])= \bigvee _{i\in I}\phi ([A_i])$ .

An automorphism is trivial if it is induced by a function on $\lambda $ . A standard definition to capture this concept is the following:

Definition 2.1. An automorphism $\phi :P^\lambda _\kappa \to P^\lambda _\kappa $ is trivial, if there is a $g:\lambda \to \lambda $ such that $\phi ([A])=[g^{-1}A]$ for all $A\subseteq \lambda $ .

However, we prefer to use forward images instead of inverse images, which can easily be seen to be equivalent:

Definition 2.2.

• For $f:A_0 \to \lambda $ with $A_0=^* \lambda $ , define $\pi _f:P^\lambda _\kappa \to P^\lambda _\kappa $ by $\pi _f([B]):=[f"(B\cap A_0)]$ for all $B\subseteq \lambda $ .
• f is an almost permutation, if there are $A_0=^*\lambda $ and $B_0=^*\lambda $ with $f:A_0\to B_0$ bijective.

(Such a $\pi _f$ is always a well-defined function.)

Lemma 2.3. Let $\phi :P^\lambda _\kappa \to P^\lambda _\kappa $ be a function. The following are equivalent $:$

(1) $\phi $ is a trivial automorphism.
(2) There is an almost permutation f such that $\phi =\pi _f$ .
(3) (Assuming $\kappa>\aleph _0$ .) There is a bijection $f:\lambda \to \lambda $ such that $\phi =\pi _f$ .

Proof (1) implies (2): Assume $\phi $ is a trivial automorphism, witnessed by g.

Then $X:=g"\lambda =^*\lambda $ (as $\phi ([X])=[g^{-1}X]= [\lambda ]$ ), and $Y:=\{\alpha \in X:\, |g^{-1}\{\alpha \}|\ne 1\}=^*\emptyset $ : Otherwise, pick $y^0_\alpha \ne y^1_\alpha $ for each $\alpha \in Y$ with $g(y^0_\alpha )=g(y^1_\alpha )=\alpha $ . So $y^0_\alpha \in g^{-1}C$ iff $y^1_\alpha \in g^{-1}C$ for any $C\subseteq \lambda $ . Set $B^i:=\{y^i_\alpha :\, \alpha \in Y\}$ for $i=0,1$ and let $[C]=\phi ^{-1}([B^0])$ . So $\phi ([C])=[g^{-1}C]=[B^0]$ , i.e., almost all $y^0_\alpha $ are in $g^{-1}C$ , but then almost all $y^1_\alpha $ are in $g^{-1}C$ as well, i.e., $[B^0]=\phi ([C])\ge [B^1]$ , a contradiction as $B^0\cap B^1=\emptyset $ .

Set $A_0:=X\setminus Y$ , and $B_0:=g^{-1}A_0$ . Note that $B_0=^*\lambda $ , as $\mathbb 0=\phi (\mathbb 0)=\phi ([Y])=[g^{-1}Y]$ . So $g\restriction B_0\to A_0$ is bijective, and we can set $f:A_0\to B_0$ the inverse. Then f is an almost permutation, and $\pi =\pi _f$ .

(2) implies (1): Let $f:A_0\to B_0$ be an almost permutation, and $g:B_0\to A_0$ the inverse (and let g be defined arbitrarily on $\lambda \setminus B_0$ ). Then $\pi _f([X])=[f"(X\cap A_0)]=[g^{-1}(X)]$ . It remains to be shown that $\pi _f$ is an automorphism: $\pi _f([\emptyset ])=[f"\emptyset ]=[\emptyset ]$ ; $\pi _f([X\cap Y])= [f"(X\cap Y\cap A_0)]=[f"(X\cap A_0)\cap f"(Y\cap A_0)]$ ; and $\pi _f([\lambda \setminus X])= [f"(A_0\setminus X)]=[B_0\setminus f"X]$ .

(2) implies (3) if $\operatorname {\mathrm {cf}}(\kappa )>\aleph _0$ : This follows from the following lemma.

Lemma 2.4 ( $\kappa>\aleph _0$ ).

Let f be a $\kappa $ -almost permutation. Then there is an $S=^*\lambda $ such that $f\restriction S:\, S\rightarrow S$ is bijective.

Proof Set $X_0:=A_0=\operatorname {\mathrm {dom}}(f)$ , and $X_{i+1}:= X_{i} \cap f" X_i \cap f^{-1}X_{i}$ , and $S:=\bigcap _{i\in \omega }X_i$ .

The $X_n$ are decreasing, and $|\lambda \setminus X_n|<\kappa $ and thus $|\lambda \setminus (f" X_n)|<\kappa $ for $n<\omega $ . Accordingly, $|\lambda \setminus S|<\kappa $ . We claim that $g:=f\restriction S$ is a permutation of S. Clearly it is injective. If $\alpha \in S$ then $\alpha \in X_n$ for all $n\in \omega $ , so $f(\alpha )\in X_{n+1}$ for all n. So $g:S\to S$ . If $\alpha \in S$ , then $\alpha \in X_{n+1}$ for all n, so $f^{-1}(\alpha )$ exists and is in $X_n$ .

Remark: For $\kappa =\lambda =\omega $ , there are trivial automorphisms that are not induced by “proper” bijections $f:\omega \to \omega $ , e.g., the automorphism $\phi $ induced by the almost permutation $n\mapsto n+1$ .Footnote ⁵

We will investigate somewhere and densely trivial automorphisms. To simplify notation, we assume $\kappa =\lambda>\aleph _0$ :

Definition 2.5 ( $\lambda>\aleph _0$ regular.).

Let $\phi :P^\lambda _\lambda \to P^\lambda _\lambda $ be an automorphism.

• $\phi $ is trivial on $A\in [\lambda ]^\lambda $ , if there is an $f:A\to \lambda $ with $\phi ([B])=[f"B]$ for all $B\subseteq A$ .
• $\phi $ is somewhere trivial, if it is trivial on some $A\in [\lambda ]^{\lambda }$ .
• $\phi $ is densely trivial, if for all $A\in [\lambda ]^{\lambda }$ there is a $B\subseteq A$ of size $\lambda $ such that $\phi $ is trivial on B.

Just as before it is easy to see that we can assume f to be a full permutation:

Fact 2.6 ( $\lambda>\aleph _0$ regular.).

An automorphism $\phi :P^\lambda _\lambda \to P^\lambda _\lambda $ is trivial on $A\in [\lambda ]^\lambda $ iff there is a bijection $f:\lambda \to \lambda $ such that $\phi ([B])=[f"(B)]$ for all $B\subseteq A$ .

Lemma 2.7 ( $\lambda>\aleph _0$ regular.).

If every automorphism of $P^\lambda _\lambda $ is somewhere trivial, then every automorphism of $P^\lambda _\lambda $ is densely trivial.

Proof Assume $\pi $ is an automorphism of $P^\lambda _\lambda $ , and fix $A\in [\lambda ]^\lambda $ . If $A=^*\lambda $ and if $\pi $ is trivial on some B, then $\pi $ is trivial on $B\cap A\subseteq A$ , so we are done. So assume $A\ne ^*\lambda $ .

Pick some representative $\pi ^*:{\mathcal {P}}(\lambda ) \to {\mathcal {P}}(\lambda )$ of $\pi $ such that $\pi ^*(A)$ and $\pi ^*(\lambda \setminus A)$ partition $\lambda $ , and such that $\pi ^*(C)\subseteq \pi ^*(A)$ for every $C\subseteq A$ . Let $i: \lambda \setminus A \to A$ and $j:\pi ^*(\lambda \setminus A)\to \pi ^*(A)$ both be bijective. Let $\pi '$ map $[D]$ to $[\pi ^*(D\cap A)\cup j^{-1}\pi ^*(i"(D\setminus A))]$ . This is an automorphism of $P^\lambda _\lambda $ , so it is trivial on some $D_0$ . If $|D_0\cap A|=\lambda $ , we are done, as $\pi '$ restricted to $D_0\cap A$ is the same as $\pi $ and trivial. So assume otherwise. Then $\pi '$ is trivial on the large set $D_0\setminus A$ . Then $\pi $ is trivial on $i"(D_0\setminus A)\subseteq A$ .

3. Under MA, every automorphism is trivial for $\omega _1\le \lambda <2^{\aleph _0}$

Theorem 3.1. Assume $\aleph _0<\kappa \leq \lambda <2^{\aleph _0}$ , $\kappa $ regular, and $ {\textrm {MA}}_{(=\lambda )}(\sigma {\textrm {-centered}})$ holds. Then every automorphism of $P^\lambda _\kappa $ is trivial.

For the proof we will use that we can separate certain sets by closed sets.

A tree T is a subset of $2^{<\omega }$ such that $s\in T\cap 2^n$ and $m\le n$ implies $s\restriction m\in T$ ; for such a T we set $\lim (T)=\{\eta \in 2^\omega :\, (\forall n\in \omega )\, \eta \restriction n\in T\}$ . A subset of $2^\omega $ is closed iff it is of the form $\lim (T)$ for some tree T.

Lemma 3.2. Assume $\aleph _0<\theta \leq \lambda <2^{\aleph _0}$ , $\operatorname {\mathrm {cf}}(\theta )>\aleph _0$ , and $ {\textrm {MA}}_{(=\lambda )}(\sigma {\textrm {-centered}})$ holds. Assume $A_0, A_1$ are disjoint subsets of $2^{\omega }$ of size $\le \lambda $ ; $|A_0|\ge \theta $ . Then there is a tree $T_0$ in $2^{<\omega }$ such that $|A_0\cap \lim (T_0)|\geq \theta $ and $A_1\cap \lim (T_0)=\emptyset $ .

If additionally $|A_1|\ge \theta $ , we get an additional tree $T_1$ such that $|A_1\cap \lim (T_1)|\geq \theta $ , $A_0\cap \lim (T_1)=\emptyset $ , and $T_0\cap T_1\subseteq 2^n$ for some n.

Proof of the lemma

In the following we identify an $x\in 2^\omega $ with the according (infinite) branch b in the tree $2^{<\omega }$ . So a branch b can be in $A_0$ or in $A_1$ (or in neither; but not both, as $A_0$ and $A_1$ are disjoint).

We define a poset Q as follows: A condition $q\in Q$ is a triple $(n_q,S_q,f_q)$ , where:

• $n_q\in \omega $ ,
• $S_q$ is a tree in $2^{<\omega }$ of the following form: $S_q$ is the union of $2^{\le n_q}$ and finitely many (infinite) branches $\{b_j:\, j\in m\}$ for some $m\in \omega $ , each $b_j\in A_0\cup A_1$ , and $b_j\restriction n_q=b_k\restriction n_q$ implies ( $b_j\in A_i$ iff $b_k\in A_i$ ).

So every $s\in S_q$ with $|s|> n_q$ is either “in $A_0$ -branches” (i.e., there is one or more $b_j\in A_0$ with $s\in b_j$ ), or “in $A_1$ -branches” (but not in both).

Note that an $s\in S_q$ of length $n_q$ is either in $A_0$ -branches, or in $A_1$ -branches, or in neither (but not in both).
• $f_{q}: S_q\to 2$ such that, for $i=0,1$ , $f_{q}(s)=i$ whenever $s\in S_q$ , $|s|\ge n_q$ and s is in $A_i$ -branches.

The order on Q is the natural one: $q\le p$ if $n_q\ge n_p$ , $S_q\supseteq S_p$ , and $f_q$ extends $f_p$ .

Q is $\sigma $ -centered witnessed by $(n_q,S_q,f_q)\mapsto (n_q, f_q\restriction 2^{\le n_q})$ : If $p,q$ are in Q with $n_p=n_q=:n$ and $f_p\restriction 2^{\le n}=f_q\restriction 2^{\le n}$ , then $(n, S_p\cup S_q, f_p\cup f_q)$ is a valid condition stronger than both p and q.

For $x\in A_i$ , the set $D_x$ of conditions containing x as branch is dense: Given $p\in Q$ , let $n_q\ge n_p$ be such that all $A_{1-i}$ -branches in p split off x below $n_q$ ; set $S_q:=S_p\cup 2^{\le n_q}\cup x$ ; and set $F_q(s)=i$ for $s\in S_q\setminus S_p$ .

Similarly, for all $n\in \omega $ , the set $D^*_n$ of conditions q with $n_q\ge n$ is dense as well.

By $\textrm {MA}_{(=\lambda )}(\sigma \text {-centered})$ and $|A_i|\le \lambda $ , we can find a filter G which has nonempty intersection with each $D_x$ for $x\in A_0\cup A_1$ as well as for each $D^*_n$ . So $F:=\bigcup _{p\in G} f_p$ is a total function from $2^{<\omega }$ to $2$ ; and for all $x\in A_i$ there is an $n_x\in \omega $ such that $m\ge n_x$ implies $F(x\restriction m)=i$ .

As $|A_0|\geq \theta $ and $\operatorname {\mathrm {cf}}(\theta )>\aleph _0$ we can assume that there is an $n^*_0$ such that $n_x=n^*_0$ for $\theta $ many $x\in A_0$ . If additionally $|A_1|\geq \theta $ , we analogously get an $n^*_1$ and set $n^*:=\max (n^*_0,n^*_1)$ ; otherwise we set $n^*:=n^*_0$ . We set $T_i^*:=\{s\in 2^{<\omega }: |s|\ge n^*,\, (\forall n^*\le k\le |s|)\, F(s\restriction k)=i\}$ and generate a tree from it; i.e., we set $T_i:=T^*_i\cup \{s\restriction m:\, m<n^*, s\in T^*_i\}$ . As we have seen above, $\lim (T_i)\cap A_i\ge \theta $ for $i=0$ (and, if $|A_1|\ge \theta $ , for $i=1$ as well). Clearly $T_0\cap T_1\subseteq 2^{n^*}$ ; and $\lim (T_i)\cap A_{i-1}$ is empty, as for any $x\in A_{i-1}$ , cofinally many n satisfy $F(x\restriction n)=i-1$ .

Proof of the theorem

Fix an automorphism $\pi $ of $P^\lambda _\kappa $ represented by some $\pi ^*:{\mathcal {P}}(\lambda )\to {\mathcal {P}}(\lambda )$ , and let $\pi ^{-1*}$ represent $\pi ^{-1}$ . We have to show that $\pi $ is trivial.

Fix an injective function $\eta :\lambda \rightarrow 2^\omega $ . Set

$$\begin{align*}C_n:=\{x\in 2^\omega:\, x(n)=0\}\text{ and } \Lambda_{n}:=\eta^{-1}C_n=\{\alpha<\lambda:\eta(\alpha)(n)=0\}.\end{align*}$$

Define $\nu :\lambda \to 2^\omega $ by

$$\begin{align*}\nu(\beta)(n)=0\text{ iff }\beta\in \pi^*(\Lambda_n).\end{align*}$$

In the following, “large” means “of cardinality ${\ge }\kappa $ ”, and “small” means not large. We will show:

(*₁) $\pi ^*(\eta ^{-1}C) =^* \nu ^{-1}C$ for $C\subseteq 2^\omega $ closed.
(*₂) $Y\subseteq \lambda \text { and }|Y|\ge \kappa $ implies $|\nu "Y|\ge \kappa $ .
(*₃) If $A_0, A_1$ are disjoint subsets of $2^\omega $ , $A_0\subseteq \nu "\lambda $ large, then $\pi ^{-1*}(\nu ^{-1}A_0)\setminus \eta ^{-1} A_1$ is large.
(*₄) If $A_0, A_1$ are disjoint subsets of $2^\omega $ , $A_0\subseteq \eta "\lambda $ large, then $\pi ^*(\eta ^{-1}A_0)\setminus \nu ^{-1} A_1$ is large.

(Note that ( $*_2$ ) is the only place where we use that $\kappa $ is regular.)

Proof:

(*₁) $\pi ^*(\eta ^{-1}C_n) = \nu ^{-1}C_n$ holds by definition of $\nu $ . As $\pi $ honors ${<}\kappa $ -unions and complements, and as the $C_n$ generate the open sets, this equation (with $=^*$ ) holds whenever C is generated by ${<}\kappa $ -unions and complements from the open sets, in particular, if C is closed.
(*₂) Fix $x\in 2^\omega $ . Then $\eta ^{-1}\{x\}$ has at most one element (as $\eta $ is injective), and $\eta ^{-1} \{x\}=^* \pi ^{-1*}\nu ^{-1}\{x\}$ by ( $*_1$ ). That is, $\nu ^{-1}\{x\}$ is small. And $Y\subseteq \bigcup _{x\in \nu "Y}\nu ^{-1}\{x\}$ , so as $\kappa $ is regular we get $|\nu "Y|\ge \kappa $ .)
(*₃) Using the previous lemma (with $\kappa $ as $\theta $ ) we get a tree $T_0$ separating $A_0$ and $A_1$ . That is, $\lim (T_0)\cap A_1=\emptyset $ and $X:=\lim (T_0)\cap A_0$ is large. As $X\subseteq A_0\subseteq \nu "\lambda $ , we get that $\nu ^{-1} X$ is large. And $\nu ^{-1} X=\nu ^{-1}\lim (T_0)\cap \nu ^{-1}A_0=^*\pi ^*(\eta ^{-1}\lim (T_0))\cap \nu ^{-1}A_0$ , the last equation by ( $*_1$ ). This implies $\eta ^{-1}\lim (T_0)\cap \pi ^{-1*}(\nu ^{-1}A_0)$ is large, and so $\pi ^{-1*}(\nu ^{-1}A_0)\setminus \eta ^{-1} A_1$ is large.
(*₄) We get an analogous result when interchanging $\nu $ and $\eta $ and using $\pi ^*$ instead of $\pi ^{-1*}$ .

We claim that the following sets $N_i$ are all small:

(1) $N_{1}:=\{\alpha \in \lambda :\, (\lnot \exists \beta \in \lambda ) \,\eta (\alpha )=\nu (\beta )\}$ .
(2) $N_{2}:=\{\alpha \in \lambda :\, (\exists ^{(\geq 2)} \beta \in \lambda )\, \eta (\alpha )=\nu (\beta )\}$ .
(3) $N_{3}:=\{\beta \in \lambda :\, (\lnot \exists \alpha \in \lambda )\, \eta (\alpha )=\nu (\beta )\}$ .

Proof.

(3) Assume $N_{3}$ is large. Set $A_0:=\nu " N_{3}$ , which is large by ( $*_2$ ); and $A_1:=\eta "\lambda $ . So $A_0$ and $A_1$ are disjoint, and by ( $*_3$ ) $\pi ^{-1*}\nu ^{-1}A_0\setminus \eta ^{-1}A_1$ is large, but $\eta ^{-1}A_1=\lambda $ .
(1) Assume $N_{1}$ is large. Set $A_0=\eta "N_{1}$ (large, as is injective) and $A_1:=\nu "\lambda $ . So $A_0$ and $A_1$ are disjoint, and by ( $*_4$ ) $\pi ^*(\eta ^{-1}A_0)\setminus \nu ^{-1} A_1$ is large, but $\nu ^{-1} A_1=\lambda $ .
(2) Assume that $N_{2}$ is large. For every $\alpha \in N_{2}$ , let $\beta _\alpha ^0\neq \beta _\alpha ^1$ in $\lambda $ be such that $\eta (\alpha )=\nu ({\beta _\alpha ^0})=\nu ({\beta _\alpha ^1})$ . For $i\in \{0,1\}$ , set $Y_i:=\{ \beta _\alpha ^i:\ \alpha \in N_{2}\}$ and $X_i:=\pi ^{-1*}(Y_i)$ (without loss of generality disjoint), and $A_i:=\eta " X_i$ . So the $A_i$ are large and disjoint, and we can find a tree $T_0$ such that $A_0\cap \lim (T_0)$ is large, and $A_1\cap \lim (T_0)$ is empty.

As $A_0\subseteq \eta "\lambda $ , this implies that the inverse $\eta $ -image of $A_0\cap \lim (T_0)$ is also large. That is,

$\eta ^{-1}(A_0\cap \lim (T_0))=\eta ^{-1}A_0\cap \eta ^{-1}\lim (T_0)=^* X_0\cap \pi ^{-1*}\nu ^{-1}\lim (T_0)$ is large (for the last equation we use ( $*_1$ )). Therefore also $Y_0 \cap \nu ^{-1}\lim (T_0)$ is large, and so, by ( $*_2$ ), $\nu "(Y_0 \cap \nu ^{-1}\lim (T_0))=\lim (T_0)\cap \nu " Y_0 $ is large as well.

On the other hand, $\lim (T_0)\cap A_1$ is empty, so $0=^*\pi ^*\eta ^{-1} (\lim (T_0)\cap A_1)=^* \pi ^*\eta ^{-1}\lim (T_0)\cap \pi ^*\eta ^{-1}A_1$ . Using ( $*_1$ ) for $\lim (T_0)$ , and noting that $\pi ^*\eta ^{-1}A_1=Y_1$ , this set is (almost) equal to $Y_1\cap \nu ^{-1}\lim (T_0)$ which therefore is also small, and so $ \lim (T_0)\cap \nu "Y_1$ is small.

So we know that $\lim (T_0)\cap \nu " Y_0$ is large and $ \lim (T_0)\cap \nu "Y_1$ is small, but $\nu " Y_0=\nu " Y_1$ , a contradiction.

Note that this implies:

(*₅) $X\cap Y$ small implies $\nu "X\cap \nu "Y$ small, for $X,Y\subseteq \lambda $ .
(*₆) $\nu ^{-1}\nu " X=^*X$ for $X\subseteq \lambda $ .

Proof:

(*₅) Assume otherwise. Without loss of generality we can assume that X and Y are disjoint, and by (3) that $\nu "X$ and $\nu " Y$ both are subsets of $\eta "\lambda $ . Then $\nu "X\cap \nu "Y\subseteq \eta "N_{2}$ is small.
(*₆) Set $Y:=\nu ^{-1}\nu "X\setminus X$ . Then $\nu "Y\subseteq N_2\cup N_3$ is small, and by ( $*_2$ ) Y is small.

Set $D:=\lambda \setminus (N_{1}\cup N_{2})$ and define $e:D\to \lambda $ such that $e(\alpha )$ is the (unique) $\beta \in \lambda $ with $\eta (\alpha )=\nu (\beta )$ . Clearly e is injective. We claim that e generates $\pi $ , i.e., that the following are small (where we can assume $X\subseteq D$ ):

(4) $N_{4}:=\pi ^*(X)\setminus e"X$ .
(5) $N_{5}:=e"X\setminus \pi ^*(X)$ .

Proof.

(4) Assume that $N_{4}$ is large. Set $Y=\pi ^{-1*}(N_{4})$ , without loss of generality $Y\subseteq X$ and $\pi ^*(Y)=N_{4}$ . So $\pi ^*(Y)$ is disjoint from $e"Y$ (as it is even disjoint from $e"X$ ). We set $A_0:=\nu "\pi ^*(Y)$ and $A_1:=\nu "e"Y$ , by ( $*_5$ ) we can assume they are disjoint, and by ( $*_2$ ) both are large (e is injective).

By ( $*_3$ ), $\pi ^{-1*}(\nu ^{-1}A_0)\setminus \eta ^{-1} A_1$ is large.

$\eta ^{-1}(A_1)=Y$ , as $\nu (e(\alpha ))=\eta (\alpha )$ for all $\alpha \in D$ . And $\pi ^{-1*}(\nu ^{-1}A_0)=^*Y$ by definition and ( $*_6$ ), a contradiction.
(5) The same proof works: This time we set $Y=e^{-1}N_{5}$ ; see that $\pi ^*(Y)$ and $e"Y$ are disjoint and large; set $A_0:=\nu " \pi ^*(Y)$ and $A_1:=\nu " e" Y$ ; use ( $*_3$ ) to see that $Y\setminus \eta ^{-1}\nu " e"Y=Y\setminus Y$ is large, a contradiction.

4. For measurables, GCH implies a nontrivial automorphism

Theorem 4.1. If $\lambda $ is measurable and $2^\lambda =\lambda ^+$ , then there is a nontrivial automorphism of $P^\lambda _\lambda $ .

Proof Let $\mathcal D$ be a normal ultrafilter on $\lambda $ and denote by $\mathcal I:=[\lambda ]^\lambda \setminus \mathcal D$ its dual ideal restricted to sets of size $\lambda $ .

Since $2^\lambda =\lambda ^+$ , we can list all permutations of $\lambda $ as $\{e_\alpha : \alpha <\lambda ^+\}$ ; and analogously all elements of $\mathcal I $ as $\{X_\alpha : \alpha <\lambda ^+\}$ .

We will construct, by induction on $\alpha <\lambda ^+$ a set $A_\alpha \in \mathcal I$ and a permutation $f_\alpha $ of $A_\alpha $ , such that for $\alpha <\beta $ :

(1) $A_\alpha \subseteq ^*A_\beta $ .
(2) $X_\alpha \subseteq A_{\alpha +1}$ .
(3) $f_{\alpha }(x)=f_{\beta }(x)$ for almost all $x\in A_{\alpha }\cap A_{\beta }$ .
(4) There is some $X\subseteq A_{\alpha +1}$ of size $\lambda $ such that $e_\alpha "X$ and $f_{\alpha +1}"X$ are disjoint.

(Note that by $x\subseteq ^*y $ we mean $|y\setminus x|=\lambda $ , not $y\setminus x\in \mathcal I$ ; and the same for ‘almost all”.)

The construction:

• Successor stages $\alpha +1$ : Fix any $B\in \mathcal I$ disjoint to $A_\alpha $ such that $A_\alpha \cup B\supseteq X_\alpha $ . Set $C:=e_\alpha " B\cap A_\alpha $ .

First assume that $|C|=\lambda $ . Then set $A_{\alpha +1}=A_\alpha \cup B$ and let $f_{\alpha +1}$ extend $f_\alpha $ by the identity on B. Then (4) is witnessed by $X:=e_\alpha ^{-1}C$ .So we assume $|C|<\lambda $ . Partition B into large sets $B_0,B_1,B_2$ such that $e_\alpha "B_i$ is disjoint to $A_\alpha $ for $i=0,1$ . Set $A_{\alpha +1}:=A_\alpha \cup B\cup e_\alpha " B$ , and define $f_{\alpha +1}$ on B such that the restriction to $B_i$ is a bijection op $e_\alpha " B_{1-i}$ for $i=0,1$ , and the restriction to $B_2$ a bijection to $e"B_2\setminus A$ . Then (4) is witnessed by $X:=B_0$ .
• Limit stages $\delta $ of cofinality ${<}\lambda $ : Let $\xi := \operatorname {\mathrm {cf}}(\delta )$ and choose $\langle \alpha _i: i<\xi \rangle $ a cofinal increasing sequence converging to $\delta $ . The union $\bigcup _{i< \xi }A_{\alpha _i}$ is, by ${<}\lambda $ completeness, in $\mathcal I$ . Remove $<\lambda $ many points to get a subset $A_\delta $ such that:
1. – For all $i<j<\xi $ , $f_i$ and $f_j$ agree on $A_{\alpha _i}\cap A_\delta $ .
2. – For all $i<\xi $ , $f_i\restriction (A_{\alpha _i}\cap A_\delta )$ is a full permutation (we can do this as in Lemma 2.4).
Then $f_\delta $ , defined as the union of the $f_{\alpha _i}$ , is a permutation of $A_\delta $ and almost extends each $f_{\alpha _i}$ .
• Limit stages $\delta $ of cofinality $\lambda $ : We choose an increasing cofinal sequence $\langle \alpha _i: i<\lambda \rangle $ converging to $\delta $ . By induction on $i\in \lambda $ we construct $ A^{\prime }_i=^*A_{\alpha _i}$ , such that:
1. – $A^{\prime }_i\cap i=\emptyset $ .
2. – The $f_{\alpha _i}$ ’s fully extend each other on the $A^{\prime }_i$ ’s, i.e., if $x\in A^{\prime }_i\cap A^{\prime }_j$ then $f_{\alpha _i}(x)=f_{\alpha _j}(x)$ .
3. – $f_{\alpha _i}: A^{\prime }_i \rightarrow A^{\prime }_i$ is a “full” permutation.
We can do this by removing from $A_{\alpha _i}$ : the points less than i, the points where $f_{\alpha _i}$ disagrees with some previous $f_{\alpha _j}$ for any $j<i$ ; and by removing ${<}\lambda $ many points to get a full permutation.

Now we can set $A_\delta $ and $f_\delta $ to be the unions of $A^{\prime }_{i}$ and $f_{\alpha _i}$ , respectively, for $i<\delta $ . Note that $A_\delta $ is in $\mathcal I$ (as it is a subset of the diagonal union); and $f_\delta $ is a permutation of $A_\delta $ satisfying (3).

Note that for all $X\subseteq \lambda $ , either $X\in \mathcal I$ or $\lambda \setminus X\in \mathcal I$ (but not both), i.e., either X or $\lambda \setminus X$ is $\subseteq ^* A_\alpha $ for coboundedly many $\alpha <\lambda $ .

This allows us to define the automorphism $\pi $ as follows: For $X\in [\lambda ]^\lambda $ ,

$$\begin{align*}\pi([X]):=\begin{cases} [f_\alpha"X], &\text{ if } X\in \mathcal I, X\subseteq^* A_\alpha \text{ for some }\alpha<\lambda^+\text{ (Case 1),}\\ [\lambda \setminus f_\alpha"(\lambda\setminus X) ],&\text{ if } X\notin \mathcal I,\lambda \setminus X\subseteq^* A_\alpha \text{ for some }\alpha<\lambda^+\text{ (Case 2)}.\end{cases}\end{align*}$$

Note that in Case 2, $\pi ([X])=[(\lambda \setminus A_\alpha )\cup (A_\alpha \setminus f^{\prime \prime }_\alpha (A_\alpha \setminus X))]=[(\lambda \setminus A_\alpha )\cup f_\alpha "(X\cap A_\alpha )]$ , as $f_\alpha "A_\alpha =^*A_\alpha $ .

$\pi $ is well defined on $[\lambda ]^\lambda $ , as exactly one of X or $\lambda \setminus X$ will eventually be $\subseteq ^* A_\alpha $ .

$\pi $ is an automorphism: $\pi ([\emptyset ])=\emptyset $ . $\pi $ honors complements: If X is Case 1, then $\pi ([\lambda \setminus X])$ is by definition (Case 2) $[\lambda \setminus f_\alpha "(X)]$ . $\pi $ honors intersections $X\cap Y$ : This is clear if both sets are the same Case. Assume that X is Case 1 and Y Case 2. Then $X\cap Y\subseteq X$ is Case 1, and for any $\alpha $ suitable for both X and Y we have

$$\begin{align*}\pi([X])\wedge \pi([Y])& = [f_\alpha" X\cap ((\lambda\setminus A_\alpha)\cup f_\alpha"(Y\cap A_\alpha))]\\ &= [f_\alpha" X \cap f^{\prime\prime}_\alpha(Y\cap A_\alpha)]= [f_\alpha" (X\cap Y))]. \end{align*}$$

$\pi $ is not trivial: Every automorphism e is an ${e_\alpha }$ for some $\alpha \in \lambda ^+$ ; and according to (4) there is some $X_\alpha \subseteq A_{\alpha +1}$ (and therefore in $\mathcal I$ ) of size $\lambda $ such that $e_\alpha "X_\alpha $ is disjoint to $f_{\alpha +1}" X_\alpha $ , a representative of $\pi ([X_\alpha ])$ .

5. For inaccessible $\lambda $ , all automorphisms can be densely trivial

In this section, we always assume the following (in the ground model):

Assumption 5.1. $\lambda $ is inaccessible and $2^\lambda =\lambda ^+$ . We set $\mu :=\lambda ^{++}$ .

In the rest of the paper, we will show the following:

Theorem 5.2. $(\lambda $ is inaccessible and $2^\lambda =\lambda ^+.)$ There is a $\lambda $ -proper, ${<}\lambda $ -closed, $\lambda ^{++}$ -cc poset P (in particular, preserving all cofinalities) that forces: $2^{\lambda }=\lambda ^{++}$ , and every automorphism of $P^\lambda _\lambda $ is densely trivial.

By Lemma 2.7, it is enough to show that every automorphism is somewhere trivial.

5.1. The single forcing $\boldsymbol Q$

Definition 5.3. We fix a strictly increasing sequence $(\theta ^*_\zeta )_{\zeta <\lambda }$ with $\theta ^*_\zeta <\lambda $ regular and $\theta ^*_\zeta>2^{|\zeta |}$ .

• Let $(I^*_{\zeta })_{\zeta \in \lambda }$ be an increasing interval partition of $\lambda $ such that $I^*_\zeta $ has size $2^{\theta ^*_\zeta }$ ; and fix a bijection of $I^*_\zeta $ and $2^{\theta ^*_\zeta }$ . Using this (unnamed) bijection, we set $[s]:=\{\ell \in I^*_\zeta : \ell>s\}$ for $s\in 2^{<\theta ^*_\zeta }$ .

So the $[s]$ are cones, i.e., the set of all branches in $I^*_\zeta $ extending s.

For $\zeta <\lambda $ , we set $I^*({<}\zeta ):=\bigcup _{\ell <\zeta } I^*_\ell $ , and analogously $I^*({\le }\zeta ):=I^*({<}\zeta +1)$ , $I^*({\ge }\zeta ):=\lambda \setminus I^*({<}\zeta )$ , and $I^*({\ge }\zeta ,{<}\xi ):=I^*({\ge }\zeta )\cap I^*({<}\xi )$ .
• A condition q of the forcing notion Q is a function with domain $\lambda $ such that, for all $\zeta \in \lambda $ , $q(\zeta )$ is a partial function from $I^*_\zeta $ to $2$ , and such that for a club-set $C^q\subseteq \lambda $ :
1. – if $\zeta \notin C^q$ , then $q(\zeta )$ is total,
2. – otherwise, the domain of $q(\zeta )$ is $I^*_\zeta \setminus [s^q_\zeta ]$ for some $s^q_\zeta \in 2^{<\theta ^*_\zeta }$ .
$C^q$ and $s^q_\zeta $ are uniquely determined by q; and q is uniquely determined by the partial function $\eta ^q:\lambda \to 2$ defined as $\bigcup _{\zeta \in \lambda }q(\zeta )$ .
• q is stronger than p if $\eta ^{q}$ extends $\eta ^p$ .

(This implies that $C^{q}\subseteq C^{p}$ , and that $s^{q}_\zeta $ extends $s^p_\zeta $ for all $\zeta \in C^{q}$ .)

The following is straightforward:

Lemma 5.4. Q has size $2^{\lambda }$ , is ${<}\lambda $ -closed, and adds a generic real in $2^\lambda $ .

Proof ${<}\lambda $ -closure is obvious, but for later reference we would like to point out the “problematic cases”:

Let $(p_i)_{i<\delta }$ be decreasing for a limit ordinal $\delta <\lambda $ .

As a first approximation, set $\eta ^*:=\bigcup _{i<\delta }\eta ^{p_i}$ (a partial function) and $C^*:=\bigcap _{i<\delta }C^{p_i}$ (a club set) and $s^*_\zeta :=\bigcup _{i<\delta }s^{p_i}_\zeta \in 2^{\le \theta ^*_\zeta }$ for $s\in C^*$ . For $\zeta \notin C^*$ , $\eta ^*$ is indeed total on $I^*_\zeta $ , and for $\zeta \in C^*$ the domain in $I^*_\zeta $ is $I^*_\zeta \setminus [s^*_\zeta ]$ .

The problematic case is when $s^*_\zeta $ is unbounded in $\theta ^*_\zeta $ . (This can only happen if $\operatorname {\mathrm {cf}}(\delta )= \theta ^*_\zeta $ , in particular for at most one $\zeta $ .) In this case we can just pick any extension $\eta ^q$ of $\eta ^{*}$ by filling all values in $I^*_{\le \zeta }$ . This gives the desired q, with $C^{q_\delta }=C^*\setminus {\zeta +1}$ .

Remarks.

• The limits of ${<}\lambda $ -sequences of conditions are not “canonical” if there are problematic $\zeta $ ’s, as we have to fill in arbitrary values.
• determines the generic filter, by . This follows from the following facts:
1. – p and q are compatible (as conditions in Q) iff $\eta ^p$ and $\eta ^q$ are compatible as partial functions and $X_{p,q}:=\{\zeta \in C^p:\, s^p_\zeta \text { and }s^q_\zeta \text { are incomparable}\}$ is non-stationary.
2. – If $p,q$ are such that $X_{p,q}$ is stationary, then the set of conditions r such that $\eta ^r$ and $\eta ^q$ are incompatible (as partial functions) is dense below p.

5.2. Properness of $\boldsymbol Q$ : Fusion and pure decision

Definition 5.5. We say $q\le _{\xi } p$ , if $q\le p$ , $\xi \in C^q$ , and $q\restriction \xi =p\restriction \xi $ .

$q\le ^+_{\xi } p$ means $q\le _\xi p$ and $q(\xi )=p(\xi )$ .

(Note the difference between $q\le ^+_{\xi } p$ and $q\le _{\xi +1} p$ : The former does not require $\xi +1\in C^q$ .)

Lemma 5.6. Let $\delta \le \lambda $ be a limit ordinal, $\xi \in \lambda $ , and $(q_i)_{i<\delta }$ a sequence in Q.

(1) If $\delta <\lambda $ and $q_{j}<^+_{\xi } q_i$ for all $i<j<\delta $ , then there is a $q_\infty $ such that $q_\infty <^+_{\xi }q_i$ for all i.
(2) If $q_{j}<_{\xi _i} q_i$ for $i<j<\delta $ , where $(\xi _i)_{i\in \delta }$ is a strictly increasingFootnote ⁶ sequence in $\lambda $ , then there is a $($ canonical) limit $q_\infty $ such that $q_\infty <_{\xi _i}q_i$ for all i.

Proof (1): We perform the same construction as in the proof of Lemma 5.4. If there is a problematic case $\zeta $ , then $\zeta>\xi $ (as for $\zeta '\le \xi $ the conditions $q_i(\zeta ')$ are constant). We can then make $\eta ^*$ total on $I^*(>\xi ,\le \zeta )$ . (It may not be enough to make it total on $I^*_\zeta $ , as $C^*\setminus \{\zeta \}$ might not be club.)

(2): Define $q_\infty (\zeta ):= \bigcup _{i\in \delta } q_i(\zeta )$ for $\zeta \in \lambda $ .

This is a non-total function (on $I^*_\zeta $ ) iff $\zeta \in C^{q_\infty }:=\bigcap _{i<\delta } C^{q_i}$ , which is closed as intersection of closed sets, and also unbounded: If $\delta <\lambda $ because we have a small intersections of clubs, if $\delta =\lambda $ as it contains each $\xi _i$ .

There are no problematic cases: If $\zeta $ is below some $\xi _i$ , then $q_j(\zeta )$ is eventually constant. If $\zeta $ is above all $\xi _i$ , which can only happen if $\delta <\lambda $ , then $\operatorname {\mathrm {cf}}(\delta )\le \delta \le \sup (\xi _i) \le \zeta < \theta ^*_\zeta $ .

So Q satisfies fusion; and we will now show that it also satisfies “pure decision”; standard arguments then imply that Q is $\lambda $ -proper and $\lambda ^\lambda $ -bounding.

Definition 5.7. Let $\xi \in \lambda $ , $q\in Q$ .

• $\operatorname {\mathrm {POSS}}^{Q}(\xi ):=2^{I^*({<}\xi )}$ . So in the extension $V[G]$ , for each $\xi $ there will be exactly one $x\in \operatorname {\mathrm {POSS}}^{Q}(\xi )$ compatible with (or equivalently: an initial segment of) the generic real . We write “” or “G chooses x” for this x.
• $\operatorname {\mathrm {poss}}(q,\xi )$ is the set of $x\in \operatorname {\mathrm {POSS}}^{Q}(\xi )$ compatible with $ \eta ^q$ (as partial functions), or equivalently: $x\in \operatorname {\mathrm {poss}}(q,\xi )$ iff . So q forces that exactly one $x\in \operatorname {\mathrm {poss}}(q,\xi )$ is chosen by G.
• Let be a name for an ordinal. We say that $q \xi $ -decides , if there is for all $x\in \operatorname {\mathrm {poss}}(q,\xi )$ an ordinal $\tau ^x$ such that q forces .

Note that for $p\in Q$ and $\zeta \in C^p$ , $q\le ^+_{\zeta } p$ is equivalent to $\operatorname {\mathrm {poss}}(q,\zeta +1)=\operatorname {\mathrm {poss}}(p,\zeta +1)$ , while $q\le _\zeta p$ is equivalent to $\zeta \in C^q$ and $\operatorname {\mathrm {poss}}(q,\zeta )=\operatorname {\mathrm {poss}}(p,\zeta )$ .

Lemma 5.8. Assume $p\in Q$ , $\zeta \in C^p$ , $x\in \operatorname {\mathrm {poss}}(p,\zeta +1)$ , and $r\le p$ extendsFootnote ⁷ x. Then there is a $q\le ^+_\zeta p$ forcing: . This condition is denoted by $r\vee (p\restriction \zeta +1)$ .

Proof We set $q(\ell )$ to be $p(\ell )$ for $\ell \le \zeta $ , and $r(\ell )$ otherwise. If $q'\le q$ forces then $q'$ extends x and thus $q'\le r$ .

Corollary 5.9. $($ “Pure decision”) Let be a name for an ordinal, $p\in Q$ , and $\zeta \in C^p$ . Then there is a $q\le ^+_\zeta p$ which $(\zeta +1)$ -decides .

Proof Let $(x_i)_{i\in \delta }$ enumerate $\operatorname {\mathrm {poss}}(p,\zeta +1)$ , for some $\delta <\lambda $ . Set $p_0=p$ , and define a $\le ^+_\zeta $ -decreasing sequence $p_j$ by induction on $j\le \delta $ : For limits use Lemma 5.6(1), and for successors choose some $r\le p_i$ deciding with a stem extending $x_i$ and set $p_{i+1}$ to $r\vee p_i\restriction (\zeta +1)$ .

From fusion and pure decision we get bounding and $\lambda $ -proper, via “continuous reading of names.” This is a standard argument, and we will not give it here; we will anyway prove a more “general” variant (for an iteration of Q’s), in Lemmas 5.25 and 5.27.

Fact 5.10.

• Q has continuous reading of names: If is a Q-name for a $\lambda $ -sequence of ordinals, and $p\in Q$ , then there is a $q\le p$ and there are $\xi _i\in \lambda $ such that $q \xi _i$ -decides for all $i\in \lambda $ .
• Q is $\lambda ^\lambda $ -bounding. That is, for every name and $p\in Q$ there is an $f\in \lambda ^\lambda $ and $q\le p$ such that q forces for all $i\in \lambda $ .
• Q is $\lambda $ -proper. This means: If N is a ${<}\lambda $ -closed elementary submodel of $H(\chi )$ of size $\lambda $ containing Q, with $\chi $ sufficiently large and regular, and if $p\in Q\cap N$ , then there is a $q\le p N$ -generic (i.e., forcing that each name of an ordinal which is in N is evaluated to an ordinal in N).

For completeness, we also mention the following well-known fact (the proof is straightforward):

Fact 5.11. Assume $\kappa $ is regular, and that the forcing notion R is $\kappa ^\kappa $ -bounding. Then R preserves the regularity of $\kappa $ , and every club-subset of $\kappa $ in the extension contains a ground model club-set.

5.3. The iteration $\boldsymbol P$

Let us first recall some well-known facts:

Facts 5.12. A ${<}\lambda $ -closed forcing preserves cofinalities ${\le }\lambda $ and also the inaccessibility of $\lambda $ . The ${\le }\lambda $ -support iteration of ${<}\lambda $ -closed forcings is ${<}\lambda $ -closed.

We will iterate the forcings Q from the previous section in a ${<}\lambda $ -closed ${\le }\lambda $ -support iteration of length $\mu :=\lambda ^{++}$ :

Definition 5.13. Let $(P_\alpha ,Q_\alpha )_{\alpha < \mu }$ be the ${\le }\lambda $ -support iteration such that each $Q_\alpha $ is the forcing Q (evaluated in the $P_\alpha $ -extension). We will write P to denote the limit.

Remark. One way to see that P is proper is to use the framework of [Reference Rosł anowski and Shelah6]. However, we will need an explicit form of continuous reading for P anyway, which in turn gives properness for free.

Definition 5.14. Assume that $w\in [\mu ]^{<\lambda }$ and $\xi \in \lambda $ .

• is the sequence of $Q_\alpha $ -generic reals added by P.
• $\operatorname {\mathrm {POSS}}(w,\xi ):=2^{w\times I^*({<}\xi )}$ . Exactly one $x\in \operatorname {\mathrm {POSS}}(w,\xi )$ is extended by , we write “x is selected by G,” or “ $x\triangleleft G$ .”
• $\operatorname {\mathrm {poss}}(p,w,\xi ):=\{x\in \operatorname {\mathrm {POSS}}(w,\xi ):\, \lnot p\Vdash \lnot x\triangleleft G\}$ .
• Let be a name of an ordinal. is $(w,\xi )$ -decided by q, if there are $(\tau ^x)_{x\in \operatorname {\mathrm {poss}}(q,w,\xi )}$ such that q forces .

Clearly, if is $(w,\xi )$ -decided by q, and if $q'\le q$ , $w'\supseteq w$ and $\xi '\ge \xi $ , then is $(w',\xi ')$ -decided by $q'$ .

Remark. If $q\in P (w,\zeta )$ -decides some $P_\alpha $ -name , then the same q will generally not $(w\cap \alpha ,\xi )$ -decide for any $\xi $ .Footnote ⁸

In the following, whenever we say that $q (w,\zeta )$ -decides something, we implicitly assume that $w\in [\mu ]^{<\lambda }$ and $\zeta \in \lambda $ .

Definition 5.15. Let be a P-name for a $\lambda $ -sequence of ordinals.

• q continuously reads , if there are $(w_i,\xi _i)_{i\in \lambda }$ such that $q (w_i,\xi _i)$ -decides for each $i\in \lambda $ .
• P has continuous reading, if for each such and $p\in P$ there is some $q\le p$ continuously reading .

The following is a straightforward standard argument:

Fact 5.16. If P has continuous reading, then it is $\lambda ^\lambda $ -bounding.

As a first step towards pure decision, let us generalize the $\le _\zeta $ -notation we defined for Q:

Definition 5.17. Let $p\in P$ , $w\in [\mu ]^{<\lambda }$ , and $\xi \in \lambda $ .

• p fits $(w,\xi )$ , if $w\subseteq \operatorname {\mathrm {dom}}(p)$ and $p\restriction \alpha \Vdash \xi \in C^{p(\alpha )}$ for all $\alpha \in w$ .
• $q\leq _{w,\xi } p$ means: $q\leq p$ , and for all $\alpha \in w$ , $q\restriction \alpha $ forces $q(\alpha )<_\xi p(\alpha )$ .
• $q\le ^+_{w,\xi } p$ is defined analogously using $<^+_\xi $ instead of $<_\xi $ .

Obviously $q\le ^+_{w,\xi } p$ implies $q\leq _{w,\xi } p$ ; and $q\leq _{w,\xi } p$ implies that both p and q fit $(w,\xi )$ .

Remark. In contrast to the single forcing (or a product of such forcings), $q\le _{w,\xi }p$ (or $q\le ^+_{w,\xi }p$ ) does not imply $\operatorname {\mathrm {poss}}(q,w,\xi )=\operatorname {\mathrm {poss}}(p,w,\xi )$ .Footnote ⁹ More explicitly, setting $w=\{0,1\}$ , it is possible that $x\in \operatorname {\mathrm {poss}}(p,w,\xi )$ but p does not force that implies $x(1)\in \operatorname {\mathrm {poss}}(p(1),\xi )$ . (But see Section 5.5.)

5.4. Continuous reading and properness of $\boldsymbol P$

Lemma 5.18. If $q_i$ is a $\leq ^+_{w,\zeta }$ -decreasing sequence of length $\delta <\lambda $ , then there is an $r\leq ^+_{w,\zeta }q_i$ for all $i<\delta $ .

Proof Set $\operatorname {\mathrm {dom}}(r):=\bigcup _{i\in \delta }\operatorname {\mathrm {dom}}(q_i)$ , without loss of generality closed under limits. By induction on $\alpha \in \operatorname {\mathrm {dom}}(r)$ we know that $r\restriction \alpha \le q_i\restriction \alpha $ for all i, and define $r(\alpha )$ as follows: If $\alpha \in w$ , we know that the $q_i(\alpha )$ are $\le ^+_\zeta $ -increasing. Using Lemma 5.6(1), we pick some $r(\alpha )$ such that $r(\alpha )\le ^+_\zeta q_i(\alpha )$ for all i. If $\alpha \notin w$ , we just pick any $r(\alpha )\le q_i(\alpha )$ for all i.

It is easy to see that P satisfies a version of fusion:

Lemma 5.19. Assume $(p_i)_{i<\delta }$ is a sequence of length $\delta \le \lambda $ , such that $p_{j}\leq _{w_i,\xi _i} p_i$ for $i\leq j<\delta $ , $w_i\in [\mu ]^{<\lambda }$ increasing, $\xi _i\in \lambda $ strictly increasing. Set $w_\infty :=\bigcup _{i<\delta } w_i$ , $\operatorname {\mathrm {dom}}_\infty :=\bigcup _{i<\delta }\operatorname {\mathrm {dom}}(p_i)$ , and $\xi _\infty :=\sup _{i<\delta }\xi _i$ . If $\delta =\lambda $ , we additionally assume $w_\infty =\operatorname {\mathrm {dom}}_\infty $ .

Then there is a limit $q_\infty $ with $\operatorname {\mathrm {dom}}(q_\infty )=\operatorname {\mathrm {dom}}_\infty $ such that $q_\infty \le _{w_i,\xi _i} p_i$ for all $i<\delta $ .

If $\delta <\lambda $ , then $q_\infty $ fits $(w_\infty ,\xi _\infty )$ .

(If $w_\infty =\operatorname {\mathrm {dom}}_\infty $ , then the limit $q_\infty $ is “canonical”.)

Proof We define $q_\infty (\alpha )$ by induction on $\operatorname {\mathrm {dom}}_\infty $ . We assume that we already have $q':=q_\infty \restriction \alpha $ which satisfies $q'\le _{w_i\cap \alpha , \xi _i}p_i$ for all $i<\delta $ .

Case 1: $\alpha \notin w_\infty $ (this can only happen if $\delta <\lambda $ ): We know that $q'$ forces that $(p_i(\alpha ))_{i<\delta }$ is a decreasing sequence, and we just pick some $q_\infty (\alpha )$ stronger then all of them.

Case 2: $\alpha \in w_\infty $ : Let $i^*$ be minimal such that $\alpha \in w_{i^*}$ . We know that $q'$ forces for all $i^*\le i<j<\delta $ that $p_j(\alpha )<_{\xi _i}p_i(\alpha )$ , so according to Lemma 5.6(2) there is a limit $q_\infty (\alpha )<_{\zeta _i}p_i(\alpha )$ (so in particular $q'\Vdash \zeta _i\in C^{q_\infty (\alpha )}$ for all $i\ge i^*$ ).

Now assume $\delta <\lambda $ . If $\alpha \in w_\infty $ , then it is in $w_i$ for coboundedly many $i<\delta $ . In other words, $p_j\restriction \alpha \Vdash \zeta _i\in C^{p_j(\alpha )}$ for coboundedly many $i\in \delta $ and all $j>i$ , which implies $q_\infty \restriction \alpha \Vdash \xi _\infty \in C^{q_\infty (\alpha )}$ .

Preliminary Lemma 5.20. Let p fit $(w,\zeta )$ , $x\in \operatorname {\mathrm {poss}}(p,w,\zeta +1)$ , and let $r\le p$ extend x, i.e., $r\Vdash x\triangleleft G$ . Then there is a $q\le ^+_{w,\zeta } p$ forcing that $x\triangleleft G$ implies $r\in G$ .

Proof Set $\operatorname {\mathrm {dom}}(q):=\operatorname {\mathrm {dom}}(r)$ . We define $q(\alpha )$ by induction on $\alpha \in \operatorname {\mathrm {dom}}(q)$ and show inductively:

• $q\restriction \alpha \le ^+_{w\cap \alpha ,\zeta }p\restriction \alpha $ .
• $q\restriction \alpha \Vdash (x\restriction \alpha \triangleleft G_\alpha \rightarrow r\restriction \alpha \in G_\alpha )$ .

For notational convenience, we assume $\operatorname {\mathrm {dom}}(p)=\operatorname {\mathrm {dom}}(r)$ (by setting $p(\alpha )=\mathbb {1}_Q$ for any $\alpha $ outside the original domain of p).

Assume we already have constructed $q_0=q\restriction \alpha $ . Work in the $P_\alpha $ -extension $V[G_\alpha ]$ with $q_0\in G$ .

Case 1: $r\restriction \alpha \notin G_\alpha $ . Set $q(\alpha ):=p(\alpha )$ .

Case 2: $r\restriction \alpha \in G_\alpha $ . Then $r(\alpha )\le p(\alpha )$ . If $\alpha \notin w$ , we set $q(\alpha ):=r(\alpha )$ ; otherwise we set $q(\alpha )$ to be $r(\alpha )\vee (p(\alpha )\restriction \zeta +1)$ as in Lemma 5.8.

If $\alpha \in w$ , then in both cases we get $q\restriction \alpha \vDash q(\alpha )\le ^+_\zeta p(\alpha )$ . Also, if $G_{\alpha +1}$ selects $x\restriction (\alpha +1)$ , then at stage $\alpha $ we used, by induction, Case 2; so then $r(\alpha )\in G(\alpha )$ as .

We can iterate the construction for all elements of $\operatorname {\mathrm {poss}}(w, \zeta +1)$ , which gives us:

Lemma 5.21. If p fits $(w,\zeta )$ and is a name for an ordinal, then there is a $q\le ^+_{w,\zeta }p$ which $(w,\zeta +1)$ -decides .

Proof We enumerate $\operatorname {\mathrm {poss}}(p,w,\zeta +1)$ as $(x_i)_{i\in \delta }$ . We start with $p_0:=p$ . Inductively we construct $p_\ell $ : If at step $\ell $ , if $x_\ell $ is not in $\operatorname {\mathrm {poss}}(p_\ell ,w,\zeta +1)$ any more, then we set $p_{\ell +1}:=p_\ell $ . Otherwise, pick an $r\le p_\ell $ that decides to be some $\tau ^{x_\ell }$ and extends $x_\ell $ . Then apply 5.20 to get $p_{\ell +1}\le ^+_{w,\zeta } p_\ell $ which forces that $x_\ell \triangleleft G$ implies . At limits use Lemma 5.18.

For the proof of Lemma 5.23 we will need a variant where the “height” $\zeta $ is not the same for all elements of w, more specifically:

Preliminary Lemma 5.22. Assume that p fits $(w,\zeta )$ and $p\restriction \alpha ^*\Vdash \zeta ^*\in C^{p(\alpha ^*)}$ , and that is a name for an ordinal. Then there is a $q\le ^+_{w,\zeta }p$ such that $q\restriction \alpha ^*\Vdash q(\alpha ^*)\le ^+_{\zeta ^*}p(\alpha ^*)$ and there is a (ground model) set A of size ${<}\lambda $ such that .

Proof This is just a notational variation of the previous proof. For notational simplicity we assume $\alpha ^*\notin w$ .

First we have to modify Preliminary Lemma 5.20: A candidate is a pair $(x,a)$ where $x\in \operatorname {\mathrm {POSS}}(w,\zeta )$ and $a^*\in \operatorname {\mathrm {POSS}}^{Q}(\zeta ^*)$ . Assume that $(x,a)$ is a candidate, that $p\in P$ fits $(w,\zeta )$ , and that $p\restriction \alpha ^*\Vdash \zeta ^*\in C^{p(\alpha ^*)}$ , and assume that $r\leq p$ extends $(x,a)$ , i.e., . Then there is a q such that

(*)

The same proof works, with the obvious modifications:

When defining $q(\alpha )$ , we inductively show:

• $q\restriction \alpha \le ^+_{w\cap \alpha ,\zeta }p\restriction \alpha $ and if $\alpha>\alpha ^*$ then $q\restriction \alpha ^*\Vdash q(\alpha ^*)\le ^+_{\zeta ^*} p(\alpha ^*)$ ,
• , unless $\alpha \le \alpha ^*$ in which case we omit the clause about $\alpha ^*$ .

Again, in the $P_\alpha $ -extension we have:

Case 1: $r\restriction \alpha \notin G_\alpha $ . Set $q(\alpha ):=p(\alpha )$ .

Case 2: $r\restriction \alpha \in G_\alpha $ . Then $r(\alpha )\le p(\alpha )$ . If $\alpha \notin w\cup \{\alpha ^*\}$ , we set $q(\alpha ):=r(\alpha )$ ; otherwise we set $q(\alpha )$ to be $r(\alpha )\vee (p(\alpha )\restriction \zeta +1)$ as in Lemma 5.8.

Then we can show (*) as before.

We then enumerate all candidates (there are ${<}\lambda $ many) as $(x_\ell ,a_\ell )$ , and at step $\ell $ , if $(x_\ell ,a_\ell )$ is compatible with $p_\ell $ , use (*) to decide to be some .

We will now show that P is $\lambda ^\lambda $ -bounding and proper. We first give two preliminary lemmas that assume this is already the case for all $P_{\beta '}$ with $\beta '<\beta $ .

Preliminary Lemma 5.23. Let $\beta \le \mu $ , and assume that $P_{\beta '}$ is $\lambda ^\lambda $ -bounding for all $\beta '<\beta $ .

Assume $p\in P_\beta $ fits $(w,\zeta )$ , $\tilde C\subseteq \lambda $ is club, and $\alpha ^*<\beta $ .

Then there is a $q\le ^+_{w,\zeta } p$ and a $\xi \in \tilde C$ such that q fits $(w\cup \{\alpha ^*\},\xi )$ .

If additionally $\alpha ^*\in \operatorname {\mathrm {dom}}(p)$ and $p\restriction \alpha ^*\Vdash \zeta ^*\in C^{p(\alpha ^*)}$ for some $\zeta ^*\in \lambda $ , then we can additionally get $q\restriction \alpha ^*\Vdash q(\alpha ^*)\le ^+_{\zeta ^*} p(\alpha ^*)$ .

Proof For notational simplicity assume $\alpha ^*\notin w$ and $\min (\tilde C)>\max (\zeta ,\zeta ^*)$ . By induction on $\alpha \le \beta $ we show that the result holds for all $w,\alpha ^*$ with $w\cup \{\alpha ^*\}\subseteq \alpha $ .

Successor case $\alpha +1$ : Set $w_0:=w\cap \alpha $ .

By our assumption $P_\alpha $ is $\lambda ^\lambda $ -bounding, so every club-set in the $P_\alpha $ -extension contains a ground-model club (see Fact 5.11). In particular, $C^{p(\alpha )}$ contains some ground-model $C^*$ . By Lemma 5.21 (or Preliminary Lemma 5.22, if $\alpha ^*<\alpha $ ) there is a $p'\le ^+_{w_0,\zeta }p\restriction \alpha $ (also dealing with $\alpha ^*$ , if $\alpha ^*<\alpha $ ) leaving only ${<}\lambda $ many possibilities for $C^*$ . So we can intersect them all, resulting in $C'$ . Set $C":=C'\cap \tilde C$ . Apply the induction hypothesis in $P_\alpha $ to get $q'\le ^+_{w_0,\zeta } p'$ and $\xi $ in $C"$ such that $q'$ fits $(w_0,\xi )$ (and also $(\{\alpha ^*\},\xi )$ , if $\alpha ^*<\alpha $ ). Set $q:=q'\cup \{(\alpha , p(\alpha ))\}$ , so trivially $q\le ^+_{w,\zeta } p$ (and, if $\alpha =\alpha ^*$ , then $q\restriction \alpha \Vdash q(\alpha )\le ^+_{\zeta ^*} p(\alpha )$ ), and q fits $(w\cup \{\alpha \},\xi )$ .

Limit case: If w is bounded in $\alpha $ there is nothing to do. So assume w is cofinal.

Set $\alpha _0:=\min (w\setminus \alpha ^*)$ and $w_0:=(w\cap \alpha _0)\cup \{\alpha ^*\}$ . Use the induction hypothesis in $P_{\alpha _0}$ using $(p\restriction \alpha _0,w_0,\zeta ,\alpha ^*,\zeta ^*)$ as $(p,w,\zeta ,\alpha ^*,\zeta ^*)$ . This gives us some $p_0'\le ^+_{w\cap \alpha _0,\zeta } p\restriction \alpha _0$ fitting $(w_0,\zeta _0)$ and dealing with $\alpha ^*$ , for some $\zeta _0\in \tilde C$ . Set $p_0:=p'\wedge p$ .

Enumerate $w\setminus w_0$ increasingly as $(\alpha _i)_{i<\delta }$ , and set $w_j:=w_0\cup \{\alpha _i:\ i<j\}$ for $j\le \delta $ .

We will construct $p^{\prime }_i$ in $P_{\alpha _i}$ and $(\zeta _i)_{i<\delta }$ a strictly increasing sequence in $\tilde C$ , and we set $p_j:=p^{\prime }_j\wedge p$ and will get: $p_\ell $ fits $(w_\ell ,\zeta _\ell )$ ,and $p_\ell \le ^+_{w_i,\zeta _i} p_i$ for all $i<\ell \le j$ .

For successors $\ell =i+1$ , we use the induction hypothesis in $P_{\alpha _{i+1}}$ , using $(p_i\restriction \alpha _{i+1},w_i,\zeta _i,\alpha _i,\zeta )$ as $(p,w,\zeta ,\alpha ^*,\zeta ^*)$ . This gives us $p^{\prime }_{i+1}\le ^+_{w_i,\zeta _i} p_i\restriction \alpha _{i+1}$ and some $\zeta _{i+1}>\zeta _i$ in $\tilde C$ such that $p_{i+1}$ fits $(w_{i+1},\zeta _{i+1})$ and $p_{i+1}\restriction \alpha _i\Vdash p_{i+1}(\alpha _i)\le ^+_{\zeta } p_i(\alpha _i)$ .

For j limit, we set $\zeta _j:=\sup _{i<j}\zeta _i$ (which is in $\tilde C$ ), and let $p_j$ be a limit of the $(p_i)_{i<j}$ . That is, $\operatorname {\mathrm {dom}}(p_j)=\bigcup _{i<j} \operatorname {\mathrm {dom}}(p_i)$ , and for $\beta \in \operatorname {\mathrm {dom}}(p_j)$ let $p_j(\beta )$ be as follows: If $\beta \notin w$ , fix some condition $p_j(\beta )$ stronger than all $p_i(\beta )$ . Otherwise, there is a minimal $i_0<j$ such that $\beta \in w_{i_0}$ , and $p_\ell (\beta )<^+_{\zeta _i} p_i(\beta )$ for all $i_0\le i<\ell <j$ . In that case let $p_j(\beta )$ be the (canonical) limit of the $(p_i(\beta ))_{i_0\le i<j}$ , and note that $\zeta _j\in C^{p_j(\beta )}$ .

Preliminary Lemma 5.24. Let $\beta \le \mu $ , and assume that $P_{\beta '}$ is $\lambda ^\lambda $ -bounding for all $\beta '<\beta $ .

Assume that $p\in P_\beta $ fits $(w,\zeta )$ , and is a $P_\beta $ -name for a $\lambda $ -sequence of ordinals. Then there is a $q\le ^+_{w,\zeta } p$ continuously reading .

Proof Set $p_0:=p$ , $\zeta _0:=\zeta $ , $w_0:=w$ . We construct by induction on $i< \lambda p^{\prime }_i$ , $p_i$ , $\zeta _i$ , $\alpha _i$ , and $w_i$ as follows:

• Given $p_j$ , $w_j$ , and $\zeta _j$ , pick $\alpha _j\in \operatorname {\mathrm {dom}}(p_j)\setminus w_j$ by bookkeeping (so that in the end the domains of all conditions will be covered).
• Successor $j=i+1$ : Set $w_{i+1}:=w_i\cup \{\alpha _i\}$ . Find $p^{\prime }_{i+1}\le ^+_{w_i,\zeta _i} p_i$ and $\zeta _{i+1}>\zeta _i$ such that $p^{\prime }_{i+1}$ fits $(w_{i+1},\zeta _{i+1})$ (using the previous preliminary lemma).
• Limit j: Let $p^{\prime }_j$ be the canonical limit of the $(p_i)_{i<j}$ , $\zeta _j:=\sup _{i<j}(\zeta _i)$ , and $w_j:=\bigcup _{i<j} w_i$ . Note that $p^{\prime }_j$ fits $(w_j,\zeta _j)$ .
• In any case, given $p^{\prime }_j$ we pick some $p_j\le ^+_{w_j,\zeta _j} p^{\prime }_j$ which $(w_j,\zeta _j+1)$ -decides .

Then the limit q of the $p_i$ continuously reads .

Lemma 5.25. P has continuous reading (and in particular is $\lambda ^\lambda $ -bounding).

Proof Assume by induction that $P_{\beta '}$ is $\lambda ^\lambda $ -bounding for all $\beta <\beta '$ . Then the previous lemma gives us that $P_\beta $ has continuous reading of names, and thus is $\lambda ^\lambda $ -bounding.

The same construction shows $\lambda $ -properness:

Definition 5.26. Let $\chi \gg \mu $ be sufficiently large and regular. An “elementary model” is an $M\preceq H(\chi )$ of size $\lambda $ which is ${<}\lambda $ -closed and contains $\lambda $ and $\mu $ (and thus P).

Lemma 5.27. If M is an elementary model containing $p\in P$ , then there is a $q\le p$ which is strongly M-generic in the following sense $:$ For each P-name in M for an ordinal, $q (w,\zeta )$ -decides via a decision function in M (so in particular ).

(The decision function being in M is equivalent to $w\subseteq M$ , as M is ${<}\lambda $ closed.)

Proof Let be a sequence of all P-names for ordinals that are in M. Starting with $p_0\in M$ , perform the successor step of the previous construction within M; as M is closed the limits at steps ${<}\lambda $ are in M as well. Then the $\lambda $ -limit is M-generic.

5.5. Canonical conditions

We will use conditions that “continuously read themselves.”

Definition 5.28. $p\in P$ is $(w,\zeta )$ -canonical if p fits $(w,\zeta )$ and $p(\alpha )\restriction (\zeta +1)$ is $(w\cap \alpha ,\zeta +1)$ -decided by $p\restriction \alpha $ for all $\alpha \in w$ .

Facts 5.29. Let p be canonical for $(w,\zeta )$ .

(1) If $q\le ^+_{w,\zeta } p$ , then q is canonical for $(w,\zeta )$ and $\operatorname {\mathrm {poss}}(p,w,\zeta +1)=\operatorname {\mathrm {poss}}(q,w,\zeta +1)$ .
(2) Let $x\in \operatorname {\mathrm {poss}}(p,w,\zeta +1)$ . There is a naturally defined $p\wedge x\le p$ such that $p\Vdash (p\wedge x\in G \leftrightarrow x\triangleleft G)$ . $\{p\wedge x:\, x\in \operatorname {\mathrm {poss}}(p,w,\zeta +1)\}$ is a maximal antichain below p.
(3) Let $x\in \operatorname {\mathrm {poss}}(p,w,\zeta +1)$ . In an intermediate $P_\alpha $ -extension $V[G_\alpha ]$ with $x\restriction \alpha \triangleleft G_\alpha $ the rest of x, i.e., $x\restriction [\alpha ,\mu ]$ , is compatible with $p/G_\alpha $ in the quotient forcing.

Or equivalently: If $r_0\leq p\restriction \alpha $ in $P_\alpha $ extends $x\restriction \alpha $ , then there is an $r\le r_0$ extending x.

Definition 5.30. Assume $p\in P$ , and is a P-name for a $\lambda $ -sequence of ordinals. Let $E\subseteq \lambda $ be a club-set and $\bar w=(w_\zeta )_{\zeta \in E}$ an increasing sequence in $[\mu ]^{<\lambda }$ .

p canonically reads as witnessed by $\bar w$ if the following holds:

• $\operatorname {\mathrm {dom}}(p)=\bigcup _{\zeta \in E} w_\zeta $ .
• p is $(w_\zeta ,\zeta )$ -canonical for all $\zeta \in E$ .
• $p\restriction \alpha \Vdash C^{p(\alpha )}=E\setminus (\zeta ^{\prime }_\alpha )$ for some (ground model) $\zeta ^{\prime }_\alpha $ .
• is $(w_\zeta ,\zeta +1)$ -decided by p for all $\zeta \in E$ .

If $\sigma $ is the constant $0$ sequence (or any sequence in V), we just say “p is canonical” (as witnessed by $\bar w$ ).

Lemma 5.31. For p, as above, there is a $q\le p$ canonically reading .

If $p\in P_\alpha $ and is a $P_\alpha $ -name for some $\alpha <\mu $ , then $q\in P_\alpha $ .

Proof We just have to slightly modify the proof of Lemma 5.24.

We will construct $p_j$ , $\xi _j$ , and $\alpha _j$ by induction on $j\in \lambda $ , setting $w_j:=\{\alpha _i:i<j\}$ , such that for $0<j<k$ the following holds:

• $p_{k}\le ^+_{w_j,\xi _j} p_j$ .
• $p_{j}$ is $(w_j,\xi _j)$ -canonical.
• $p_{j} (w_j,\xi _j+1)$ -decides .
• In $p_{k}$ , for $\alpha _j\in w_k$ , $\{\zeta _i:\, j<i<k\}$ is (forced to be) an initial segment of $C^{p_k(\alpha _j)}$ .
• The $\alpha _j$ are chosen (by some book-keeping) so that $\{\alpha _i:\, i\in \lambda \}=\bigcup _{i\in \lambda }\operatorname {\mathrm {dom}}(p_i)$ .

Then the limit of the $p_j$ is as required, with $E=\{\xi _i:\, i\in \lambda \}$ and, for $\zeta =\xi _j$ in E, we use $w_j$ as $w_\zeta $ .

Set $p_0\le p$ such that $|\operatorname {\mathrm {dom}}(p_0)|=\lambda $ , and set $\xi _0:=0$ . Assume we already have $p_i,\alpha _i$ for $i<j$ (so we also have $w_j$ ).

• For j limit, let s be a limit of $(p_i)_{i<j}$ , and set $\xi _j:= \sup _{i<j}\xi _i$ . Note that s fits $(w_j,\xi )$ .
• Successor case $j=i+1$ : Find $s_0\le ^+_{w_i,\xi _i} p_i$ and $\xi _{j}>\xi _i$ such that s fits $(w_{j},\xi _{j})$ . (As in Lemma 5.23. Recall that $w_j=w_i\cup \{\alpha _i\}$ .)Strengthen $s_0$ to $s\le ^+_{w_i,\xi _i}$ so that:
1. – s still fits $(w_{j},\xi _{j})$ ,
2. – the trunk at $\alpha _i$ has length $\xi _j$ , i.e., $s\restriction \alpha _i\Vdash \min (C^{s(\alpha _i)})=\xi _j$ ),
3. – for $\alpha _{i'}$ , $i'<i$ , there are no elements in $C^{s(\alpha _{i'})}$ between $\xi _i$ and $\xi _j$ .
• Construct $s^{*}\restriction \alpha $ by recursion on $\alpha \in w_j$ , such that $s^{*}\restriction \alpha \le ^+_{w_j\cap \alpha ,\xi _j} s\restriction \alpha $ and $s^{*}\restriction \alpha (w_j\cap \alpha ,\xi _j+1)$ -decides $s(\alpha )\restriction (\xi _j+1)$ (which is the same as $s^*(\alpha )\restriction (\xi _j+1)$ ). This gives $s^{*}\le ^+_{w_j,\xi _j} s$ .
• Find $p_j\le ^+_{w_j,\xi _j} s^{*}$ which $(w_j,\xi _j+1)$ decides .
• Choose $\alpha _j\in \operatorname {\mathrm {dom}}(p_j)\setminus w_j$ by bookkeeping.

Facts 5.32.

(1) If a $P_\beta $ -name is continuously read (by some $P_\beta $ -condition p), and $\operatorname {\mathrm {cf}}(\beta )>\lambda $ , then there is an $\alpha <\beta $ such that: $p\in P_\alpha $ , and is already a $P_\alpha $ -name (formally: there is a $P_\alpha $ -name such that ).
(2) There are at most $|\alpha |^\lambda \le \lambda ^+$ many pairsFootnote ¹⁰ such that p canonically reads in $P_\alpha $ .

5.6. $\mathbf \Delta $ systems

In this section we define $\Delta $ -systems and show that such systems exist, which we will in the indirect proofs of Lemmas 5.39 and 5.54.

In Section 5.10 we will then fix a specific $\Delta $ -system for the rest of the paper.

From now on, we assume that $p_*$ forces

(5.33)

and we set, for $\beta \in \mu $ ,

where, as usual, we identify with .

Note that, other than , is a priori not a $P_{\beta +1}$ -name (but see Section 5.9).

We also fix a P-name for a representation of the inverse automorphism . Abusing notation, we call it .

With $S^\mu _{\lambda ^+}$ we denote the stationary subset of $\mu $ consisting of ordinals with cofinality $\lambda ^+$ .

Definition 5.34. Let $S\subseteq S^{\mu }_{\lambda ^+}$ be stationary, $\chi \gg \mu $ sufficiently large, and regular, and $z\in H(\chi )$ . “An elementary S-system” (using parameter z) is a sequence $(M_\beta ,p_\beta )_{\beta \in S}$ such that, for each $\beta \in S$ , $M_\beta $ is an elementary model (as in Definition 5.26) and contains z, $\beta $ , $p_*$ , , and , and $p_\beta \in P\cap M_\beta $ canonically reads witnessed by some $(w^{p_\beta }_\zeta )_{\zeta \in E^{p_\beta }}$ , which $E^{p_\beta }\subseteq \lambda $ club (cf. Definition 5.30).

By a simple $\Delta $ -system argument we can make an S-system homogeneous:

Definition 5.35. $(M_\beta , p_\beta )_{\beta \in S}$ forms a “ $\Delta $ -system,” if $\bar M,\bar p$ is an elementary S-system with parameter z, and is homogeneous in the following sense: For $\beta $ and $\beta _1< \beta _2$ in S, we get:

(1) $M_{\beta _1}\cap M_{\beta _2}\cap \mu $ is constant. We call this set the “heart” and, abusing notation, denote it with $\Delta $ . Obviously $\Delta \supseteq \lambda $ , $\Delta \supseteq \operatorname {\mathrm {dom}}(p_*)$ , $\lambda ^+\in \Delta $ , etc.
(2) $M_{\beta }\cap {\beta }=\Delta $ . So in particular $\beta $ is the minimal element of $M_\beta $ above $\Delta $ . All the non-heart elements of $M_{\beta _2}$ are above all elements of $M_{\beta _1}$ . That is, $\sup (M_{\beta _1}\cap \mu )<\beta _2$ .
(3) There is an $\in $ -isomorphism $h^*_{\beta _1,\beta _2}: M_{\beta _1}\to M_{\beta _2}$ , mapping $\beta _1$ to $\beta _2$ , $p_{\beta _1}$ to $p_{\beta _2}$ , to and fixing as well as each $\alpha $ in $\Delta $ .

Note that this implies that the continuous reading of works the same way for all $\beta $ . In particular the $E^{p_{\beta }}$ are that same E for all $\beta $ ; and if $F^\beta _\zeta $ is the function mapping $\operatorname {\mathrm {POSS}}(w^{p_\beta }_\zeta ,\zeta +1)$ to the value of (for $\zeta \in E$ ), then $h^*_{\beta _1,\beta _2}(F^{\beta _1}_\zeta )=F^{\beta _2}_\zeta $ and in particular $h^*_{\beta _1,\beta _2}(w^{p_{\beta _1}}_\zeta )=w^{p_{\beta _2}}_\zeta $ ; i.e., they are the same apart from shifting coordinates above $\Delta $ .

Lemma 5.36. Assume $S\subseteq S^\mu _{\lambda ^+}$ is stationary.

• For every $z\in H(\chi )$ and $(p^{\prime }_\beta )_{\beta \in S}$ there are $M_\beta $ and $p_\beta \le p^{\prime }_\beta $ such that $\bar M,\bar p$ is an S-system with parameter z.
• If $\bar M,\bar p$ is an S-system then there is an $S'\subseteq S$ stationary such that $(M_\beta ,p_\beta )_{\beta \in S'}$ is a $\Delta $ -system on $S'$ .

Proof The first item is trivial, using the fact that everything can be read canonically.

Using $2^\lambda =\lambda ^+$ , a standard $\Delta $ -system argument (or: Fodor’s Lemma argument) lets us thin out S to some $S^2$ so that $(M_\beta \cap \mu )_{\beta \in S^2}$ satisfies (1–3). For $\beta \in S^2$ let $\iota _\beta :M_\beta \cup \{M_\beta \}\to H(\lambda ^+)$ be the transitive collapse, and assign to $\beta $ the tuple of the $\iota _\beta $ -images of the following objects:

• $M_\beta $ , $p_\beta $ , , $\mu $ , , , and $E^{p_\beta }$ .
• For $\zeta \in E^{p_\beta }$ , the object $w^{p_\beta }_\zeta $ .
• For $\zeta \in E^{p_\beta }$ and $\gamma \in w^{p_\beta }_\zeta $ , the object $F^{p_\beta }_\gamma $ .

Again, there are $|H(\lambda ^+)|^\lambda <\mu $ many possibilities, so the objects are constant on a stationary $S'\subseteq S^2$ .

For $\alpha <\beta $ in $S'$ , we define $h^*_{\beta _1,\beta _2}:= \iota ^{-1}_{\beta _2}\circ \iota _{\beta _1}$ . (Note that $\iota _{\beta _1}(\alpha )=\iota _{\beta _2}(\alpha )$ for $\alpha \in \Delta $ .)

So in particular if we have a $\Delta $ -system on S, then $p_\beta \restriction \sup (\Delta )=p_\beta \restriction \beta \in M_\beta $ is the same for all $\beta \in S$ , and outside of $\Delta $ the domains of the $p_\beta $ are disjoint for $\beta \in S$ . In particular we get:

Fact 5.37. For a $\Delta $ -system with domain S, and $A\subseteq S$ of size ${\le }\lambda $ , the union of the $(p_{\beta })_{\beta \in A}$ is a condition in P (and stronger than each $p_\beta $ ).

Whenever $r \in P_\beta \cap M_\beta $ (as is the case for $r=p_\beta \restriction \beta $ ), we know that $r\in P_\alpha $ for $\alpha \in \Delta $ (as $M_\beta $ knows that $\beta $ has cofinality $\lambda ^+$ ).

Instead of “ $r\in P_\alpha $ for some $\alpha \in \Delta $ ” we will sometimes just state the weaker but shorter $r\in P_{\sup (\Delta )}$ .

Remark. This is an important effect also for some names. Generally, a $P_\beta $ -name in $M_\beta $ is of course not a $P_\alpha $ -name for any $\alpha <\beta $ (just take the $P_\beta $ -generic filter $G_\beta $ ). However, as we will explicitly state in Lemma 5.42, such names for subsets of $\lambda $ are, modulo some condition, $P_\alpha $ -names for some $\alpha \in \Delta $ and independent of $\beta $ . In the specific case of the $P_\beta $ -name $p_\beta (\beta )$ we do not have to increase the condition:

Definition and Lemma 5.38. $\tilde p:=p_\beta (\beta )$ is a $P_{\sup (\Delta )}$ -name independent of $\beta \in S$ .

Proof $p_\beta (\beta )\restriction \zeta +1$ is $(w^{p_\beta }_\zeta ,\zeta +1)$ -determined for cofinally many $\zeta \in E$ , where $w^{p_\beta }_\zeta \in [\beta ]^{<\lambda }$ is a subset of $M_\beta $ . So $w^{p_\beta }_\zeta \subseteq \Delta $ , and the isomorphisms between the $M_\beta $ guarantee that each $w^{p_\beta }_\zeta $ is the same, and that $p_\beta (\beta )\restriction \zeta +1$ is decided the same way. So $\tilde p$ is a $P_\gamma $ -name for $\gamma =\sup (w^{p_\beta }_\zeta )_{\zeta \in E}$ . This $\gamma $ is independent of $\beta \in S$ , and is in $\Delta $ . So $\tilde p$ is actually a $P_\alpha $ -name for some $\alpha \in \Delta $ ; and certainly a $P_{\sup (\Delta )}$ -name.

For later reference we note:

Lemma 5.39. For all but non-stationary many $\beta $ , $p_*$ forces .

(Here, $V_\beta $ denotes the $P_\beta $ -extension of the ground model.)

Proof Assume that $p_\beta \le p_*$ forces that for a $P_\beta $ -name for all $\beta \in S^*$ stationary. We can also assume that $p_\beta $ canonically reads . Pick $M_\beta $ containing $p_\beta $ and $S\subseteq S^*$ such that $(M_\beta ,p_\beta )_{\beta \in S}$ is a $\Delta $ -system, where we can assume (or get from homogeneity) that . So the are $P_\beta $ -names in $M_\beta $ and therefore $P_{\sup (\Delta )}$ -names, and are the same for all $\beta $ . Choose $\beta _1>\beta _0$ in S. So $p_{\beta _0}\wedge p_{\beta _1}$ force that , which contradicts the injectivity of and the fact that .

5.7. Preservation of cofinalities, catching canonical names

Corollary 5.40. P is $\lambda ^{++}$ -cc and preserves all cofinalities.

Proof Cofinalities ${\le }\lambda $ are preserved as P is ${<}\lambda $ -closed.

Cofinality $\lambda ^+$ is preserved by properness: Assume that it is forced by p that $\kappa $ has a cofinal $\lambda $ -sequence . Then there is an elementary model M containing p and . If $q\le p$ is M-generic, and G a P-generic filter containing q, then for all $i<\lambda $ , so $M\cap \kappa $ is a cofinal subset of $\kappa $ of size $\lambda $ in the ground model.

Cofinality $\ge \lambda ^{++}$ is preserved as P has the $\lambda ^{++}$ -cc, which we have shown in a very roundabout way with the fact about $\Delta $ -systems: If $(p^{\prime }_\alpha )_{\alpha \in \mu }$ are arbitrary conditions, then $(M_\beta ,p_\beta )$ form a $\Delta $ -system from some $p_\beta <p^{\prime }_\beta $ and stationary S, and any two (in fact, ${\le }\lambda $ many) $p_\beta $ are compatible for $\beta \in S$ .

Remark 5.41. This shows that P is $(\mu ,\lambda )$ -Knaster, i.e., for every $A\in [P]^\mu $ there is a $B\in [A]^\mu $ which is $\lambda $ -linked.

The $\lambda ^{++}$ -cc also implies: For every name for a subset of $\lambda $ (or of $\lambda ^+$ ) there is a $\beta <\mu $ and a $P_\beta $ -name such that the empty condition forces that .

Given $\alpha <\mu $ , there are ${<}\mu $ many pairs where p canonically reads in $P_\alpha $ (see Fact 5.32(2)). So there is a $g(\alpha )<\mu $ such that for each such , both and are equivalent (modulo the empty condition) to some $P_{g(\alpha )}$ -name. Let $C^*\subseteq \mu $ be the club set with $(\zeta \in C^*\, \&\, \alpha <\zeta )\ \rightarrow \ g(\alpha )<\zeta $ .

Given a $\Delta $ -system on S we can restrict it to a $\Delta $ -system on $S\cap C^*$ ; so we will assume from now on that each $\Delta $ -system we consider satisfies $S\subseteq C^*$ .

To summarize:

Lemma 5.42.

(1) If $\beta \in S$ , $p\in P_\beta $ , and a $P_\beta $ -name for a subset of $\lambda $ , then there is an $\alpha <\beta $ and a $q\le p$ canonically reading , , as $P_\alpha $ -names.

More explicitly: There is a $P_\alpha $ -name which is canonically read by q such that . (And analogously for and instead of .)
(2) If additionally $p\le p_\beta \restriction \beta $ in $P_\beta $ and , then we can additionally get: , and are $P_\alpha $ -names in $M_\beta $ independent of $\beta \in S$ .

More explicitly: Let be as above (for ). Then $\alpha \in \Delta $ , q and are in $M_\beta $ , and if $\beta '\in S$ and $h:=h^*_{\beta ,\beta '}$ , then h acts as identity on $\alpha $ , q, and , and ( $M_{\beta '}$ knows that) . (And analogously for and instead of )

Proof (1): Use Lemma 5.31 to get a $q_1\in P_\beta $ canonically reading . And if $\beta \in S$ then $\operatorname {\mathrm {cf}}(\beta )=\lambda ^+$ , so $\operatorname {\mathrm {dom}}(p)$ is bounded by some $\alpha '<\beta $ and, by Fact 5.32(1), $q_1\in P_{\alpha _1}$ for some $\alpha '\le \alpha _1<\beta $ . As $\beta \in C^*$ , and are $P_\beta $ -names. So repeat the same argument to get $q\le q_1$ in $P_\alpha $ canonically reading all three subsets of $\lambda $ .

(2): Apply (1) inside $M_\beta $ . As $\alpha \in \beta \cap M_\beta $ , we get $\alpha \in \Delta $ . As q canonically reads itself as well as , we know that h does not change q and . As h is an isomorphism, we know that $h(q)=q$ forces that .

5.8. Majority decisions

For any $(a_1,a_2,a_3)$ with $a_i\in \{0,1\}$ there is a $b\in \{0,1\}$ such that $b=a_i$ for at least two $i\in \{1,2,3\}$ . We write $b=\operatorname {\mathrm {major}}_{i=1,2,3}(a_i)$ .

Similarly, if $f_1$ , $f_2$ , $f_3$ are functions $A\to 2$ we write $\operatorname {\mathrm {major}}_{i=1,2,3}(f_i)$ for the function $A\to 2$ that maps $\ell $ to $\operatorname {\mathrm {major}}_{i=1,2,3}(f_i(\ell ))$ .

The following is a central point of the whole construction:

Lemma 5.43. Let $(M_\alpha ,p_\alpha )_{\alpha \in S}$ be a $\Delta $ -system. Pick $\beta _0<\beta _1<\beta _2<\beta _3$ in S.

(1) $p_*$ forces: If , then .
(2) Let $s=\bigwedge _{i<4} p_{\beta _i}$ . Recall that $s(\beta _i)$ is the same $P_{\sup (\Delta )}$ -name called $\tilde p$ for all i. We can strengthen s by strengthening, for $i=1,2,3$ , the condition $s(\beta _i)=\tilde p$ to some $P_{\beta _0+1}$ -names $r_i\le \tilde p$ (without changing $C^{\tilde p}$ ) such that the resulting condition forces .

(We do not have to strengthen $s(\beta _0)$ for this, i.e., we can use $r_0:=\tilde p.$ )

We describe this by “ $(r_i)_{i<4}$ honors majority.”

Recall that $\nu _1=^*\nu _2$ denotes that $\nu _1(\ell )=\nu _2(\ell )$ for all but ${<}\lambda $ many $\ell \in \lambda $ .

Proof (1) Identifying $2^\lambda $ with $P(\lambda )$ , we have $\operatorname {\mathrm {major}}_{i=1,2,3}{f_i}= (f_1\cap f_2)\cup (f_2\cap f_3)\cup (f_1\cap f_3)$ for any tuple $(f_i)_{i=1,2,3}$ . As represents an automorphism, we get . Apply this to .

(2) Work in the $P_{\beta _0+1}$ -extension. Recall $\tilde p:=p_{\beta _0}(\beta _0)$ . So both $\tilde p$ and are already determined, and extends $\eta ^{\tilde p}$ . Set $r_0:=\tilde p$ .

Set $s_1:=(0,0)$ , $s_2:=(0,1)$ , $s_3:=(1,0)$ . For $\zeta \in C^{\tilde p}$ and $i=1,2,3$ , we define $r_i(\zeta )\supseteq \tilde p(\zeta )$ as follows:

(5.44)

So $\eta ^{r_i}$ agrees on its domain with , and each $\ell \in \lambda $ is in $\operatorname {\mathrm {dom}}(\eta ^{r_i})$ for at least two $i\in \{1,2,3\}$ . Accordingly, an extension by a generic filter G with $r_i\in G(\beta _i)$ for all $i<4$ will satisfy . (We do not even have to assume that any $p_\beta \in G$ .)

Remark 5.45. Let $p^{\prime }_{\beta _1}$ be the condition where we strengthen $p_{\beta _1}(\beta _1)$ to $r_1$ . Note that $p^{\prime }_{\beta _1}$ is not in $M_{\beta _1}$ , as $\beta _0\notin M_{\beta _1}$ and $r_1$ is defined using . Similarly (basically the same): $r_1[G_{\beta _1}]\notin M_{\beta _1}[G_{\beta _1}]$ , even if we assume that $G_{\beta _1}$ is $M_{\beta _1}$ -generic. But generally we will not be interested in $M_\beta $ -generic conditions or extensions (we needed generic conditions only in Lemma 5.27, which in turn is needed for Corollary 5.40). And while usually most conditions we consider can be constructed within (and therefore will be elements of) some $M_\beta $ , this is generally not required (an example are the $s_i$ ’s in the following lemma).

The same proof works if we do not start with the $p_\beta $ but with any stronger conditions, as long as they still “cohere” in the way that the $p_{\beta _i}$ cohere:

Lemma 5.46. Let $(M_\alpha ,p_\alpha )_{\alpha \in S}$ be a $\Delta $ -system, $\beta _0<\beta _1<\beta _2<\beta _3$ in S, and $s_{i}\le p_{\beta _i}$ for $i=0,1,2,3$ such that $:$

• $\operatorname {\mathrm {dom}}(s_i)\subseteq M_{\beta _i}$ .
• $s^*:=s_i\restriction \beta _i$ is the same for all i.
• $s^*$ forces that the $s_{i}(\beta _i)$ are the same for all i.

(In the usual sense $:$ The $s_{i}(\beta _i)$ are continuously read from generics below $\beta _0$ in the same way for each $i<4$ .)

Then there is condition stronger than all $s_{i}$ forcing that and thus .

5.9. is in the $\beta +1$ -extension

We now show that can be assumed to be a $P_\beta $ -name.

The following definitions, in particular everything concerning the notion of coherence, is used only in this section. In the rest of the paper, we will use from this section only Lemma 5.54, i.e., the fact that .

Remark. Why do we introduce this (rather annoying) notion of coherence? Well, we would like to simultaneously construct something like $s_i\le p_{\beta _i}$ where each $s_i$ ends up in $M_{\beta _i}$ . We cannot directly do this in $M_{\beta _0}$ , as $M_{\beta _0}$ does not know about, e.g., $\beta _1$ . So instead, we construct four different $s_i'\le p_{\beta _0}$ in $M_{\beta _0}$ in such a way (a “coherent” way) and use $s_i:= h^*_{\beta _0,\beta _i}(s_i')$ .

Let us for now (until Lemma 5.54) fix an arbitrary $\Delta $ -system $(M_\beta ,p_\beta )_{\beta \in S}$ as well as $\beta _0<\beta _1<\beta _2<\beta _3$ in S. For notational convenience, set

$$\begin{align*}\beta:=\beta_0. \end{align*}$$

Definition 5.47.

• $\bar q=(q_i)_{i<4}$ in $M_\beta $ is called coherent, if each $q_i$ is stronger than $p_{\beta }$ and $q_i\restriction (\beta +1)$ is the same for all $i<4$ .
• If $\bar q$ is coherent, then $\bigwedge _{i<4}h^*_{\beta ,\beta _i}(q_i)$ is a valid condition in P, and we call it $q^*$ .

I.e., $q^*$ is the union of the copies of $q_i$ in $M_{\beta _i}$ ; and the copy for $q_0$ is just $q_0$ .

$r\in P$ is called coherent, if $r=q^*$ for some coherent $\bar q\in M_{\beta }$ .

Facts.

• The $p_{\beta _i}$ are coherent, more correctly:

The condition $\bigwedge _{i\in 4}p_{\beta _i}$ is coherent; equivalently: The tuple $\big (h^{*-1}_{\beta ,\beta _i}(p_{\beta _i})\big )_{i<4}$ is coherent.
• Any coherent r is stronger than $\bigwedge _{i<4}p_{\beta _i}$ .
• If $\bar q$ is coherent, $r_i\le q_i$ in $M_{\beta }$ for $i<4$ , and $r_i\restriction \beta _i$ is the same for all $i<4$ , then $\bigwedge _{i<4}h^*_{\beta ,\beta _i}(r_i)$ is (a valid condition and) compatible with $q^*$ .
• $r\in P$ is coherent iff: $\operatorname {\mathrm {dom}}(r)\subseteq \bigcup _{i<4} M_{\beta _i}$ , $r\restriction (\mu \cap M_{\beta _i})\in M_{\beta _i}$ is stronger than $p_{\beta _i}$ , and each $r(\beta _i)$ is forced to be the same condition.

In that case, $r=q^*$ for $q_i:=h^{*-1}_{\beta ,\beta _i}(r_i)$ and $r_i:=r\restriction (\mu \cap M_{\beta _i})$ .

Lemma 5.48. If r is coherent, then it can be strengthenedFootnote ¹¹ to forceFootnote ¹² .

Proof This follows from Lemma 5.46, using $s_i:=r\restriction (\mu \cap M_{\beta _i})$ .

Definition 5.49.

• $\bar w=(w_i)_{i<4}$ is coherent, if $w_i\in [\mu ]^{<\lambda }$ is in $M_{\beta }$ and $w_i\cap (\beta +1)$ is independent of i.

In the following we always assume that $\bar q$ and $\bar w$ are coherent.
• $\bar q$ fits $(\bar w,\zeta )$ , if each $q_i$ fits $(w_i,\zeta )$ .
• $\bar q$ is $(\bar w,\zeta )$ -canonical, if each $q_i$ is $(w_i,\zeta )$ -canonical.
• $\bar r\le ^{+}_{\bar w,\zeta }\bar q$ means: $\bar r$ is coherent, and $r_i\le ^+_{w_i,\zeta }q_i$ for all $i<4$ .
• $\bar x=(x_i)_{i<4}$ is defined to be in $\operatorname {\mathrm {poss}}(\bar q,\bar w,\zeta )$ if $x_i\in \operatorname {\mathrm {poss}}(q_i,w_i,\zeta )$ and $x_i\restriction \beta $ is independent of i. Such a $\bar x$ will be called coherent possibility.

(Note that the $x_i(\beta )$ in a coherent possibility can be different for different $i<4$ . Also note that such an $\bar x$ is automatically in $M_\beta $ , which is ${<}\lambda $ -closed.)

Note that if $\bar r\le ^{+}_{\bar w,\zeta }\bar q$ and $\bar q$ is $(\bar w,\zeta )$ -canonical, then $\bar r$ and $\bar q$ have the same coherent $(\bar w,\zeta +1)$ -possibilities (see Fact 5.29(1)).

Several of the previous constructions result in coherent 4-tuples when applied to coherent 4-tuples. In particular:

Lemma 5.50.

(1) Assume $(\bar q^j)_{j\in \delta }$ is a sequence of coherent 4-tuples such that, for each $i<4$ , the i-part $(q^j_i)_{j\in \delta }$ satisfies the assumptions of Lemma 5.18.

Then for each i, the lemma (in $M_\beta $ ) gives us a limit r, which we call $q^\delta _i$ .

We can choose the $q^\delta _i$ so that they form a coherent 4-tuple.
(2) The same applies to Lemma 5.19. That is, we can get a coherent fusion limit from a $\lambda $ -sequence of coherent tuples.
(3) Assume $\bar p$ fits $(\bar w,\zeta )$ , and $\alpha _i\in \mu $ such that $w^{\prime }_i:=w_i\cup \{\alpha _i\}$ is coherent. Then there is a $\xi>\zeta $ and a $\bar q\le ^+_{\bar w,\zeta }\bar p$ which fits $(\bar w',\xi )$ and is $(\bar w',\xi )$ -canonical.
(4) Assume $\bar q$ is coherent and (for simplicity) $(\bar w,\zeta )$ -canonical with $\beta \in w_i$ (which is independent of $i<4$ ), and are names of ordinals. Then there is an $\bar r\le ^{+}_{\bar w,\zeta }\bar q$ such that is $(\bar w,\zeta +1)$ -decided by $\bar r$ .

By this we mean that is $(w_i,\zeta +1)$ -decided by $r_i$ for all $i<4$ .

Proof For the first items, we just have to look at the proofs of the according lemmas (For (3) this is Preliminary Lemmas 5.23 and 5.24) and note that coherent input gives us coherent output. In the following we will prove (4). We work in $M_\beta $ .

Enumerate all coherent possibilities as $(\bar x_k)_{k\in K}$ . Set $\bar r^0:=\bar q$ . We now construct $\bar r^{k+1}$ from $\bar r:=\bar r^k$ where we assume $\bar r^k\le ^{+}_{\bar w,\zeta }\bar q$ .

• Find $s_0$ stronger than $r_0$ and extending $x_0$ , deciding .
• $s^*:=(s_0\restriction \beta )\wedge r_1$ is stronger than $r_1$ , as $\bar r$ is coherent. Strengthen $s^*(\beta )=r_1(\beta )=r_0(\beta )$ to $s_0(\beta )$ , but replace the trunk with $x_1(\beta )$ . Then $s^*\restriction \beta $ forces that $s^*(\beta )\le r_1(\beta )$ , as $x_1\restriction \beta = x_0\restriction \beta $ and as $x_1(\beta )$ is guaranteed to be possible, because $r_1$ is canonical. Further strengthen $s^*$ (above $\beta $ ) to extend (the rest of) $x_1$ ; and then strengthen the whole condition once more to decide . Call the result $s_1$ .
• Do the same for $i=2$ , starting with $s_1$ , resulting in $s_2$ , and then for $i=3$ , starting with $s_2$ , resulting in some $s_3$ .So $s_i\le r_i$ extends $x_i$ and decides , and $s_3\restriction \beta \le s_i\restriction \beta $ and $s_3(\beta )$ is stronger than $s_i(\beta )$ “above $\zeta +1$ .”
• We define $r^{\prime }_i\le r_i$ as follows: $\operatorname {\mathrm {dom}}(r^{\prime }_i)=(\operatorname {\mathrm {dom}}(s_3)\cap \beta )\cup \operatorname {\mathrm {dom}}(s_i)$ . We define $r^{\prime }_i(\alpha )$ inductively such that $r^{\prime }_i\restriction \alpha \le ^+_{w_i\cap \alpha ,\zeta } r_i$ forces that $x_i\restriction \alpha \triangleleft G$ implies $s_i\restriction \alpha \in G$ .
1. – For $\alpha \le \beta $ :
  
  If $s_3\restriction \alpha \notin G_\alpha $ , set $r^{\prime }_i(\alpha )=r_i(\alpha )$ . Assume otherwise. So $s_3(\alpha )$ is defined and stronger than $r_i(\alpha )=r_3(\alpha )$ . If $\alpha \notin w_i$ (which implies $\alpha <\beta $ ), set $r^{\prime }_i(\alpha )=s_3(\alpha )$ . Otherwise, use $s_3(\alpha )\vee (r_3(\alpha )\restriction \zeta +1)$ , as in Lemma 5.8.
2. – For $\alpha>\beta $ , we do the same but we use $s_i$ instead of $s_3$ . In more detail:
  
  If $s_i\restriction \alpha \notin G_\alpha $ , set $r^{\prime }_i(\alpha )=r_i(\alpha )$ . Assume otherwise. If $\alpha \notin w_i$ , set $r^{\prime }_i(\alpha )=s_i(\alpha )$ . Otherwise, use $s_i(\alpha )\vee (r_i(\alpha )\restriction \zeta +1)$ .
We can use this $\bar r'$ as $\bar r^{k+1}$ : It is coherent, $\bar r'\le ^{+}_{\bar w,\zeta }\bar r^k$ , and $r^{\prime }_i$ decides assuming $x_i\triangleleft G$ .

Coherent tuples $\bar q$ naturally define a P-condition $q^*$ . However, we have to assume that $\bar q$ is canonical to guarantee that coherent $\bar q$ possibilities correspond to $q^*$ -possibilities:

Lemma 5.51. Assume $\bar q$ and $\bar w$ coherent. We set $w^*:=\bigcup _{i<4}h^*_{\beta ,\beta _i}(w_i)$ . Let $\bar x$ be in $\operatorname {\mathrm {poss}}(\bar q,\bar w,\zeta +1)$ .

(1) $\bar q$ fits $(\bar w,\zeta )$ iff $q^*$ fits $(w^*,\zeta )$ .
(2) $\bar r\le ^+_{\bar w,\zeta }\bar q$ iff $r^*\le ^+_{w^*,\zeta } q^*$ .
(3) Assume $\bar q$ fits $(\bar w,\zeta )$ . Then $\bar q$ is $(\bar w,\zeta )$ -canonical iff $q^*$ is $(w^*,\zeta )$ -canonical.
(4) Assume that $\bar q$ is $(\bar w,\zeta )$ -canonical. Let $x^*$ be the union of the $h^*_{\beta ,\beta _i}(x_i)$ . Then $x^*\in \operatorname {\mathrm {poss}}(q^*, w^*,\zeta +1)$ ; and every element of $\operatorname {\mathrm {poss}}(q^*, w^*,\zeta +1)$ is such an $x^*$ for some $\bar x\in \operatorname {\mathrm {poss}}(\bar q,\bar w,\zeta +1)$ .
(5) Assume that $\bar q$ is $(\bar w,\zeta )$ -canonical. Then $\bar q (\bar w,\zeta +1)$ -decides iff $q^* (w^*,\zeta +1)$ -decides all .

Proof Assume $\alpha \in w_i$ . Set $\alpha ':= h^*_{\beta ,\beta _i}(\alpha )\in w^*$ and $q':=h^*_{\beta ,\beta _i}(q_i)$ .

(1) Assume $q_i,\alpha $ satisfy $q_i\restriction \alpha \Vdash \zeta \in C^{q_i(\alpha )}$ . By absoluteness they satisfy it in $M_{\beta }$ , so the $h^*_{\beta ,\beta _i}$ -images $q',\alpha '$ satisfy it in $M_{\beta _i}$ , which again is absolute; and $q^*\restriction \alpha '\le q'\restriction \alpha '$ forces that $q^*(\alpha ')=q'(\alpha ')$ . For the other direction, assume (in $M_{\beta }$ ) some $s\le q_i\restriction \alpha $ forces $\zeta \notin C^{q_i(\alpha )}$ . Then $h^*_{\beta ,\beta _i}(s)$ is compatible with $q^*$ and forces $\zeta \notin C^{q^{\prime }_i(\alpha ')} =C^{q^*(\alpha ')}$ .

In the same way we can show (2), as well as (5) and the trivial directions of (3) and (4). For example, if $\bar q$ is $(\bar w,\zeta )$ -canonical, then $q^*$ is $(w^*,\zeta )$ -canonical. For this, use the fact that every element $y^*\in \operatorname {\mathrm {poss}}(q^*, w^*,\zeta +1)$ “induces” a coherent possibility $\bar y$ (which is true whether $\bar q$ is canonical or not). And if additionally $\bar x\in \operatorname {\mathrm {poss}}(\bar q, \bar w,\zeta +1)$ , then $x^*\in \operatorname {\mathrm {poss}}(q^*, w^*,\zeta +1)$ ; and if each $q_i$ forces that $x_i\triangleleft G$ implies , then $q^*$ forces that $x^*\triangleleft G$ implies .

We omit the (also straightforward) proofs of the other directions of (3) and (4) (which we do not need in this paper).

In the following, whenever we mention $q^*$ or $w^*$ , we assume $\bar w$ , $\bar q$ to be coherent and in $M_{\beta }$ . We will (and can) use $x^*$ only if $\bar q$ additionally is canonical (otherwise $x^*$ will generally not be a possibility for $q^*$ ). In this case, every P-generic filter containing $q^*$ will select an $x^*$ for some coherent possibility $\bar x$ .

Lemma 5.52. Assume $\bar q$ is coherent, are P-names in $M_\beta $ for elements of $2^\lambda $ , andFootnote ¹³ . Then there is a coherent $\bar r\le \bar q$ , and sequences $(\zeta ^j)_{j\in \lambda }$ and $(\bar w^j)_{j\in \lambda }$ such that $\bar r$ is $(\bar w^j,\zeta ^j)$ -canonical for all j, and for all $\bar x\in \operatorname {\mathrm {poss}}(\bar r, \bar w^j,\zeta ^j+1)$ there is some $\ell \in I^*({>}\zeta ^j,{<}\zeta ^{j+1})$ and $\bar b=(b_i)_{i<4}$ , with $b_i\in 2$ , violating majorityFootnote ¹⁴ such that for all $i<4$

As the $p_{\beta _i}$ are coherent, we can apply the lemma to (for all i) and get:

Corollary 5.53. If , then there is a coherent $r^*\le \bigwedge _{i<4}p_{\beta _i}$ forcing that

Proof of the lemma

We will construct (in $M_\beta $ ), by induction on $j\in \lambda $ , $\zeta ^j$ , $\bar w^j$ and $\bar r^j$ with $r^0_i=q_i$ , such that the following holds:

(1) $\bar r^j$ is coherent.
(2) $\bar w^j$ is coherent, for each $i<4$ the $w_i^j$ are increasing with j, and their union covers $\bigcup _{j\in \lambda }\operatorname {\mathrm {dom}}(r_i^j)$ .
(3) $\bar r^j$ is $(\bar w^j,\zeta ^j)$ -canonical.
(4) $\bar r^k\le ^{+}_{\bar w^j,\zeta ^j} \bar r^j$ for $j<k$ .
(5) If $\bar x\in \operatorname {\mathrm {poss}}(\bar r^j,\bar w^j,\zeta ^j+1)$ , then there is an $\ell \in I^*({>}\zeta ^j,{<}\zeta ^{j+1})$ and a $b\in 2$ such that for at least two $i_1,i_2$ in $\{1,2,3\}$ , $r^{j+1}_i$ forces that $x_i\triangleleft G$ implies
(*)

Then we take the usual fusion limits, as in Lemma 5.50(2), and are done.

For limits j, let $\bar r'$ be a (coherent) limit of $(\bar r^{j'})_{j'<j}$ , and set $\zeta ^*:=\sup _{j'<j}(\zeta ^{j'})$ and $w^*_i:=\bigcup _{j'<j} w^{j'}_i$ for each $i<4$ . Note that $\bar r'$ fits $(\bar w^*,\zeta ^*)$ . Then we can find coherent $\bar r^*\le ^+_{\bar w^*,\zeta ^*} \bar r'$ which is $(\bar w^*,\zeta ^*)$ -canonical, as in Lemma 5.50(3).

In successor cases $j=j'+1$ set $(\bar r^*,\bar w^*,\zeta ^*):=(\bar r^{j'},\bar w^{j'},\zeta ^{j'})$ .

In any case we want to construct $\bar r^j$ , $\bar w^j$ , and $\zeta ^j$ .

Enumerate $\operatorname {\mathrm {poss}}(\bar r^*,\bar w^*,\zeta ^*+1)$ as $(\bar x^k)_{k\in K}$ .

We define $\bar s^k$ for $k\le K$ , with $\bar s^0:= \bar r^*$ and, as usual, taking (coherent) limits at limits, such that:

• $\bar s^k$ is coherent.
• $\bar s^\ell \le ^+_{\bar w^*,\zeta ^* } \bar s^k$ for $k<\ell <K$ . (This implies that $\bar s^k$ is $(\bar w^*,\zeta ^*)$ -canonical.)
• There is a $\xi ^k$ and an $\ell \in I^*({>}\zeta ^*,{<}\xi ^k)$ and a $b\in 2$ such that
(**)

Assume we can construct these $\bar s^k$ , $\xi ^k$ for all $k\in K$ , then let $\bar s^K$ be again a (coherent) limit. We set $w^{j}_i:=w^*_i\cup \{\alpha _j\}$ such that $\bar w^{j}$ is coherent (and such that, by bookkeeping, all elements of $\operatorname {\mathrm {dom}}(p^j_i)$ will be eventually covered), and find some $\zeta ^{j}>\sup _{k\in K}(\xi ^k)$ and $\bar r^{j}\le ^+_{\bar w^*,\zeta ^*} r^*$ which is $(\bar w^{j},\zeta ^{j})$ -canonical, again as in Lemma 5.50(3). Then $\bar r^{j}$ , $\bar w^{j}$ and $\zeta ^{j}$ are as required.

So it remains to construct, for $k\in K$ , $\bar s^{k+1}$ and $\xi ^k$ , which we will do in the rest of the proof. Set $\bar s:= \bar s^k$ , $\bar x:=\bar x^k$ , $\bar w:=\bar w^*$ , and $\zeta :=\zeta ^*$ . Recall that $\bar s$ is $(\bar w, \zeta )$ -canonical, $\bar x\in \operatorname {\mathrm {poss}}(\bar s,\bar w,\zeta )$ , and we are looking for $\bar s^{k+1}\le ^+_{\bar w,\zeta }\bar s$ which satisfies (**) for $\bar x$ .

Set $s^{\prime }_i:=s_i\wedge x_i$ . It is enough to construct $t_i\le s^{\prime }_i$ such that:

• Both $t_i\restriction \beta $ and $t_i(\beta )\restriction (\lambda \setminus \zeta +1)$ are independent of i.
• .

Then we can define $\bar s^{k+1}$ in the usual way: $\operatorname {\mathrm {dom}}(s_i^{k+1})=\operatorname {\mathrm {dom}}(t_i)$ (and we can assume $\operatorname {\mathrm {dom}}(s_i)=\operatorname {\mathrm {dom}}(t_i)$ , by using trivial conditions). For $\alpha \in \operatorname {\mathrm {dom}}(t_i)$ , if $t_i\restriction \alpha \notin G_\alpha $ then set $s_i^{k+1}(\alpha )$ to be $s_i(\alpha )$ , otherwise $t_i(\alpha )\vee (s_i(\alpha )\restriction \zeta +1)$ if $\alpha \in w_i$ and $t_i(\alpha )$ otherwise. The resulting $\bar s^{k+1}\le ^+_{\bar w,\zeta } \bar s$ is coherent and $s^{k+1}_i$ forces that $x_i\triangleleft G$ implies $t_i\in G$ .

We have to introduce more notation: Fix $j\ne i$ , and $a\le s^{\prime }_j$ and $b\le s_i'\restriction \beta +1$ (in $P_{\beta +1}$ ) such that $b\restriction \beta \le a$ and $b\restriction \beta $ forces that $b(\beta )$ is stronger than $a(\beta )$ above $\zeta $ (i.e., $b\restriction \beta \Vdash \, (\forall \xi>\zeta )\, b(\beta )(\xi )\supseteq a(\beta )(\xi )$ ). Then we define $b^{[j]} \wedge a$ by

$$\begin{align*}(b^{[j]} \wedge a)(\alpha)(\xi)=\begin{cases} b(\alpha)(\xi), & \text{if }\alpha<\beta,\\ x_j(\beta)(\xi), & \text{if }\alpha=\beta\text{ and }\xi\le \zeta,\\ b(\beta)(\xi), & \text{if }\alpha=\beta\text{ and }\xi> \zeta,\\ a(\alpha)(\xi),& \text{otherwise.} \end{cases} \end{align*}$$

Note that $b^{[j]} \wedge a$ is stronger than a, but generally not stronger than b.

By our assumption, $q_0$ and therefore $s^{\prime }_0$ forces . So in an intermediate model $V[G_{\beta +1}]$ , there is some $\ell \in I^*({>}\zeta )$ such that $s^{\prime }_0/G_{\beta +1}$ does not decide . Back in V, fix some $b_0\le s^{\prime }_0\restriction (\beta +1)$ in $P_{\beta +1}$ which determines this $\ell $ .

Find $r^{\prime }_1\le b_0^{[1]}\wedge s^{\prime }_1$ which determines to be $j_1$ for some $j_1\in 2$ . Find $r^{\prime }_2\le (r^{\prime }_1\restriction \beta +1)^{[2]}\wedge s^{\prime }_2$ which determines to be $j_2$ ; analogously find $r^{\prime }_3\le (r^{\prime }_2\restriction \beta +1)^{[3]}\wedge s^{\prime }_3$ which determines to be some $j_3$ . Let $j\in 2$ be equal to at least two of $j_1,j_2,j_3$ .

Set $p:= (r^{\prime }_3\restriction \beta +1)^{[0]}\wedge s^{\prime }_0$ . In any $P_{\beta +1}$ -extension honoring $p\restriction \beta +1$ , is not determined by $p/G_{\beta +1}$ , i.e., there is a $t_0\le p$ forcing that .

We now set and $t_i:= (t_0\restriction \beta +1)^{[i]}\wedge r^{\prime }_i$ for $i=1,2,3$ . Note that $t_i\le r^{\prime }_i\le s^{\prime }_i$ extends $x_i$ and forces to be $1-j$ if $i=0$ and to be j for at least two i in $\{1,2,3\}$ .

We can now easily show:

Lemma 5.54. For all but non-stationary many $\beta \in S^{\mu }_{\lambda ^+}$

Proof We started in this section with an arbitrary $\Delta $ -system and showed that Corollary 5.53 and Lemma 5.48 hold for this system.

We now use a specific $\Delta $ -system:

Assume towards a contradiction that on a non-stationary set $S'$ there are $p_\beta \le p_*$ forcing . By strengthening we can assume that $p_\beta $ canonically reads . Let $M_\beta $ contain $p_\beta $ and let $S\subseteq S'$ be such that $(M_\beta ,p_\beta )_{\beta \in S}$ is a $\Delta $ -system. Fix $\beta _0<\beta _1<\beta _2<\beta _3$ in S. By Corollary 5.53 we get a coherent $\bar r$ stronger than $\bar p$ such that . This contradicts Lemma 5.48.

5.10. Fixing the $\mathbf \Delta $ -system

We now know that there is a stationary set $S^0\subseteq S^\mu _{\lambda ^+}$ such that for all $\beta \in S^0$ , is forced (by $p_*$ ) to be in $V_{\beta +1}$ but not in $V_{\beta }$ (see Lemmas 5.39 and 5.54).

For each $\beta \in S^0$ there is a $p^{\prime }_\beta \le p_*$ in P forcing that is equal to some $P_{\beta +1}$ -name, call it , and we choose $p_\beta \le p^{\prime }_\beta $ (we only have to strengthen the part below $\beta +1$ ) which canonically reads .Footnote ¹⁵

We now fix, as usual, for each $\beta \in S^0$ , some elementary model $M_\beta $ containing $p_\beta $ , and fix $S\subseteq S^0$ such that $(M_\beta ,p_\beta )_{\beta \in S}$ is a $\Delta $ -system.

So $p_{**}:=p_\beta \restriction \beta \le p_*$ is independent of $\beta \in S$ (it is a $P_\alpha $ -condition for some $\alpha \in \Delta $ , independent of $\beta \in S$ ); and is read continuously by $p_\beta \restriction \beta +1$ via $(w^{\prime }_\zeta )_{\zeta \in E'}$ for some $E'\subseteq \lambda $ club, with $w^{\prime }_\zeta \subseteq \beta +1$ . As usual, due to homogeneity $E'$ is independent of $\beta \in S$ , and the $w^{\prime }_\zeta $ are independent of $\beta $ apart from the shifting of the final coordinate $\beta $ via the mapping $h^*_{\beta _0,\beta _1}$ ; the same holds for the decision functions that map $\operatorname {\mathrm {poss}}(p_\zeta , w^{\prime }_\zeta ,\zeta +1)$ to .

Let E be the limit points of $E'$ , and set $w_\zeta :=\bigcup _{\nu <\zeta }w^{\prime }_\nu $ . Then is $(w_\xi ,\xi )$ -determined by $p_\beta $ for all $\xi \in E$ .

In the $P_\beta $ -extension, only remains undetermined, i.e., there are $f_{\xi }$ for $\xi \in E$ such that $p_\beta /G_\beta $ forces . The $f_\xi $ are canonically read from $p_\beta \restriction \beta $ in a way independent of $\beta $ (due to homogeneity).

Recall that $x\in \operatorname {\mathrm {poss}}(\tilde p,\xi )$ is equivalent to: $x\in 2^{I^*({<}\xi )}$ and x extends $\eta ^{\tilde p}\restriction I^*({<}\xi )$ . So the domain of $f_\xi $ is $\operatorname {\mathrm {poss}}(\tilde p,\xi )$ .

To summarize:

Fact 5.55. $(M_\beta ,p_\beta )_{\beta \in S}$ satisfies:

• $p_\beta \restriction \beta =:p_{**}\le p_*$ is a $P_{\sup (\Delta )}$ -condition independent of $\beta \in S$ .
• $p_\beta (\beta )=:\tilde p$ is a $P_{\sup (\Delta )}$ -name independent of $\beta \in S$ .
• There is a club-set $E\subseteq \lambda $ and, for $\xi \in E$ , $P_{\sup (\Delta )}$ -names such that for all $\beta \in S$ and $\xi \in E$
• If $\beta \in S$ , is a $P_\beta $ -name, $q\le p_{**}$ in $P_{\beta }$ , and are in $M_\beta $ , then we can find $\alpha \in \Delta $ and $p^{\prime }_{**}\le q$ in $P_\alpha $ which continuously reads , , and independentlyFootnote ¹⁶ of $\beta $ .

The last item follows from Lemma 5.42; and we will use it several times: Before Corollary 5.59 we find $p^2_{**}\le p_{**}$ to get names for U, $F_\xi $ etc. that are independent of $\beta $ ; before Lemma 5.63 we get $p^3_{**}\le p^2_{**}$ to get independent names for some unions, intersections, and -images; and finally after Corollary 5.70 we choose $q\le p^3_{**}$ to get an independent name for the generator $f_{\text {gen}}$ .

5.11. Local reading

So we know that we can determine initial segments of from initial segments of , more specifically, we can determine from for $I:=I^*({<}\xi )$ .

In this section we show that on unboundedly many disjoint intervals of the form $A:=I^*(\ge \xi ,<\nu )$ , we can read from just (without having to use the -values below A).

The following definition (the notion of candidate) is only used in this section. In the rest of the paper we only need Corollary 5.59.

In the following, we work in $V_\beta $ , the $P_\beta $ -extension $V[G_\beta ]$ where we assume $\beta \in S$ and $p_{**}\in G_\beta $ .

Definition 5.56. (In $V_{\beta }$ )

• For $A\subseteq \lambda $ and $\bar x=(x_i)_{i<4}$ , $x_i:A\to 2$ , we say $\bar x$ honors majority above $\zeta $ , if
$$\begin{align*}x_0(\ell)=\operatorname{\mathrm{major}}_{i=1,2,3}x_i(\ell)\text{ for all } \ell\in A\cap I^*({\ge}\zeta). \end{align*}$$
We say $\bar x$ honors $\tilde p$ , if each $x_i$ is compatible with $\eta ^{\tilde p}$ (as partial functions).
• $\bar x=(x_i)_{i<4}$ is a $(\zeta _0,\zeta _1)$ -candidate, (for $\zeta _0\le \zeta _1$ both in E) if the $x_i\in \operatorname {\mathrm {poss}}(\tilde p,\zeta _1)$ honor majority above $\zeta _0$ .

(As elements of $\operatorname {\mathrm {poss}}(\tilde p,\zeta _1)$ they automatically honor $\tilde p$ .)
• If $\bar x$ is a $(\zeta _0,\zeta _1)$ -candidate, we say “ $\bar y$ extends $\bar x$ ” if $\bar y$ is a $(\zeta _1,\zeta _2)$ -candidateFootnote ¹⁷ for some $\zeta _2\ge \zeta _1$ and each $y_i$ extends $x_i$ .

Equivalently, $\bar y=\bar x^\frown \bar b$ for some $\bar b$ , with $b_i:I^*({\ge }\zeta _1,{<}\zeta _2)\to 2$ , which honors both majority and $\tilde p$ .
• A $(\zeta _0,\zeta _1)$ -candidate $\bar y$ is “good,” if for every candidate $\bar z$ of height $\xi>\zeta _1$ that extends $\bar y$ we have:
(*₁) $$\begin{align} f_\xi(z_0)(\ell) =\operatorname{\mathrm{major}}_{i=1,2,3}f_\xi(z_i)(\ell)\text{ for all }\ell\in I^*({\ge}\zeta_1,{<}\xi). \end{align} $$

Preliminary Lemma 5.57. (In $V_\beta $ .) Every candidate can be extended to a good candidate.

Proof Assume otherwise, i.e., there is a $(\zeta ',\zeta _0)$ -candidate $\bar x$ which is a counterexample, which means:

(*₂)

$$\begin{align} \text{Whenever } \bar y \text{ is a } (\zeta_0,\zeta_1)\text{-candidate extending } \bar{x} \text{ then there is a}\\ \nonumber \xi>\zeta_1 \text{ and a } (\zeta_1,\xi)\text{-candidate } \bar{z} \text{ extending } \bar{y} \text{ which violates}~(*_{1}). \end{align} $$

We now construct $r_0\le \tilde p$ and, for $i=1,2,3$ , $Q_\beta $ -names $r_i\le \tilde p$ . All these conditions live on the same $C^*\subseteq E$ with $\min (C^*)=\zeta _0$ . The trunk of $r_i$ is $x_i$ .

We now construct inductively $C^*\restriction \zeta $ and $r_i\restriction \zeta $ .

Assume we have determined that $\zeta \in C^*$ and we have constructed each $r_i$ below $\zeta $ . Set $r_0(\zeta ):=\tilde p(\zeta )$ and pick $r_i(\zeta )$ as in (5.44), i.e., they have majority and leave enough freedom to form a valid condition.

We will now construct the $C^*$ -successor $\xi $ of $\zeta $ , together with $r_i$ on $I^*({>}\zeta ,{<}\xi )$ .

Enumerate all $(\zeta _0,\zeta +1)$ -candidates extending $\bar x$ as $(\bar y^k)_{k\in K}$ .

Let $\bar a^0$ be the empty $4$ -tuple and set $\xi _0:=\zeta +1$ . We will construct, for $k\in K$ , $\xi _k$ and some $\bar a^k$ that honors majority and $\tilde p$ , where $a^k_i$ has domain $I^*({\ge }\zeta +1,{<}\xi _k)$ and extends $a^j_i$ if $j<k$ .

If k is a limit, let $\bar a^x$ be the (pointwise) union of $\bar a^j$ with $j<k$ , and set $\xi _k:=\sup _{j<k}(\xi _j)$ .

Assume we already have $\bar a^j$ . Extend ${{\bar y}^j}^\frown \bar a^j$ to some candidate ${{\bar y}^j}^\frown \bar a^{j+1}$ of some height $\xi _{j+1}$ in E such that

(*₃)

$$\begin{align} {\bar{y}^j}^\frown \bar a^{j+1} \text{ violates } (*_{1}) \text{ for some } \ell\in I^*({\ge}\xi_j,{<}\xi_{j+1}). \end{align} $$

We can do this due to (*₂ ).

So in the end we get some $\xi>\zeta $ in E and $\bar b^\zeta $ with domain $I^*({>}\zeta ,{<}\xi )$ honoring majority and $\tilde p$ such that

(*₄)

$$\begin{align} & \text{for every } (\zeta_0,\zeta+1)-\text{candidate } \bar y \text{ extending } \bar x, \bar y^\frown \bar b^\zeta \text{ is a } (\zeta_0,\xi)-\text{candidate}\nonumber\\ &\text{violating } (*_{1}) \text{ for some } \ell\in I^*({>}\zeta,{<}\xi). \end{align} $$

We then define $C^*$ below $\xi +1$ by adding only $\xi $ , i.e., $\xi $ is the $C^*$ -successor of $\zeta $ . We extend the conditions $r_i$ by $b^\zeta _i$ for $i<4$ . That is, we have $\eta ^{r_i}(\ell )=b^\zeta _i(\ell )$ . This ends the construction of $r_i\le \tilde p$ .

Back in V, assume that (*₂ ) is forced by some $q'\le p_\beta \restriction \beta $ . Pick an increasing sequence $\beta _i$ ( $i<4$ ) in S. We take the union of $q'$ and the $p_{\beta _i}$ , call it s, and strengthen $s(\beta _i)=\tilde p$ to $r_i$ . The resulting condition $s'$ forces the following:

• for all $\xi \in C^*$ . This is because $s'\le p_{\beta _i}$ (cf. Fact 5.55).
• The honor majority above $\zeta _0$ . This is because for all $\zeta \in C^*$ , the $r_i(\zeta )$ are chosen as in (5.44) and therefore honor majority; and for $\zeta \in \lambda \setminus (C^*\cup \zeta _0)$ we use values $\bar b$ which honor majority.
• Accordingly, the honor majority above some $\gamma <\lambda $ (cf. Lemma 5.43(1)). Pick $\zeta _1$ such that $\sup (I^*({<}\zeta _1))>\gamma $ .
• So for all $\xi>\zeta _1$ the honor majority above $\zeta _1$ .
• Pick some $\zeta>\zeta _0,\zeta _1$ in $C^*$ with $C^*$ -successor $\xi $ . By construction of the $r_i$ , is $b^\zeta _i$ . As $r_i$ extends $x_i$ , is a $(\zeta _0,\zeta +1)$ -candidate extending $\bar x$ . So by (*₄ ), the violate (*₁ ) at some $\ell \in I^*({>\zeta },{<\xi })$ , a contradiction.

Let $U\subseteq \lambda $ be club. Set to be the odd elementsFootnote ¹⁸ of U. For with U-successor $\nu $ , set

$$\begin{align*}A^U_\xi:=I^*({\ge}\xi,{<}\nu) \end{align*}$$

Lemma 5.58. (In $V_{\beta }$ .) There is an $r_0\le \tilde p$ , a club $U\subseteq C^{r_0}\subseteq E$ , and, for , an $F_\xi :2^{A^U_\xi }\to 2^{A^U_\xi }$ such that $:$

• $r_0\wedge p_\beta /G_\beta $ forces that .
• $F_\xi $ is not constant $:$ There are, for $k=0,1$ , $z_\xi ^k$ in $\operatorname {\mathrm {poss}}(r_0,I^*({<}\nu ))$ and $\ell _\xi \in A^U_\xi $ such that $F_\xi (z_\xi ^k\restriction A^U_\xi )(\ell _\xi )=k$ . (Again, $\nu $ is the U-successor of $\xi $ .)

(Note: Only those elements of $2^{A^U_\xi }$ that are compatible with $r_0$ are relevant as arguments for $F_\xi $ .)

Proof We construct $r_i$ for $i<4$ and U iteratively; $C^{r_i}$ will be independent of i, call it C.

All $r_i$ have the same trunk as $\tilde p$ ; i.e., $\min (C)=\min (C^{\tilde p})=:\zeta _0$ and $r_i\restriction \zeta _0:=\tilde p\restriction \zeta _0$ . We also set $\min (U)=\zeta _0$ .

For all $\zeta \in C$ , we choose some $r^*_i(\zeta )$ as in (5.44), i.e., $r^*_0(\zeta )=\tilde p(\zeta )$ , and the $r^*_i(\zeta )$ for $i=1,2,3$ are such that the majority of their generics would be the $r^*_0(\zeta )$ -generic.

Assume that we already know that some $\zeta $ is in U (which is a subset of C), and that we know $r_i\restriction \zeta $ for $i<4$ .

We now construct the U-successor $\xi $ of $\zeta $ , $C\restriction [\zeta ,\xi ]$ , and $r_i(\nu )$ for $i<4$ and $\nu \in [\zeta ,\xi )$ .

• Even case: If $\zeta $ is an even element of U, we start with $r_i(\zeta ):=r_i^*(\zeta )$ , but then add a “shield,” or “isolator” above $\zeta $ : As in the previous proof, we iterate over all $\zeta +1$ -candidates $\bar y^j$ , but but in (*₃ ), instead of violating (*₁ ) for some $\ell $ , we demand that ${\bar y^j}^\frown \bar z^{j+1}$ is good. (We already know that every candidate can be extended to a good one.) Accordingly, we get some $\xi>\zeta $ and $\bar b^\zeta $ with domain $I^*({>}\zeta , {<}\xi )$ (and honoring majority and $\tilde p$ ) such that $\bar y^\frown \bar b^\zeta $ is good for every candidate $\bar y$ of height $\zeta +1$ ; i.e.:
(*^′ ₄) $$\begin{align} & \text{If } \bar z \text{ is a } (\zeta+1,\nu)\text{-candidate whose restriction to } I^*({>}\zeta, {<}\xi)\ \text{is}\ \bar b^\zeta, \text{ then the}\nonumber\\ & f_{\nu}(z_i) \text{ honor majority above } \xi. \end{align} $$
We now let this $\xi $ be the successor of $\zeta $ in both C and U (and extend each $p_i(\zeta )$ by $b_i$ ).
• Odd case: Now assume $\zeta $ is odd in U. Then we choose some $\xi>\zeta $ in $C^{\tilde p}$ large enough such that there are, for $k=0,1$ , $z^k_\xi $ in $\operatorname {\mathrm {poss}}(\tilde p,\xi )$ compatible with all the $r_0$ constructed so far, such that the $f_{\xi }(z^k_\xi )(\ell )=k$ for some $\ell> I^*({<}\zeta )$ . (Such $\xi $ and $\ell $ have to exist as is not in $V_\beta $ .)

We let C restricted to $[\zeta ,\xi ]$ be the same as $C^{\tilde p}$ , and set $r_i(\nu ):=r_i^*(\nu )$ for $\nu \in C\cap [\zeta ,\xi )$ . (For $\zeta \in [\zeta ,\xi )\setminus C$ there is no freedom left, i.e., $\tilde p(\zeta )$ is already completely determined, so the only choice for any $r\le \tilde p$ is $r(\zeta )=\tilde p(\zeta )$ .)

This ends the construction of U and of $r_i$ (for $i<4$ ).

Pick , let $\zeta $ be the U-predecessor and $\nu $ the U-successor. We have to show that we can determine (modulo $p_{\beta }$ ) from alone. (We already know that we can determine it from .)

Fix any $z^\zeta _*\in \operatorname {\mathrm {poss}}(r_0,\zeta +1)$ . Let $x_0\in \operatorname {\mathrm {poss}}(r_0,\nu )$ . In particular $x_0$ extends $b_0^\zeta $ . For $i=1,2,3$ , let $x_i$ be the copy of $x_0$ with the initial segment $x_0\restriction \xi $ replaced by $z^\zeta _*{}^\frown b_i^\zeta $ . Note that $\bar x$ is a candidate extending $\bar b^\zeta $ . Accordingly the $f_\nu (x_i)$ honor majority above $\xi $ . So we can define

$$\begin{align*}F_{\xi}(x_0\restriction A^U_\xi):=\operatorname{\mathrm{major}}_{i=1,2,3}f_{\nu}(x_i) \restriction A^U_\xi = f_\nu(x_0) \restriction A^U_\xi. \end{align*}$$

This is well-defined,Footnote ¹⁹ and $r_0\wedge p_\beta /G_\beta $ forces that .

We now summarize this lemma, which was shown in $V_\beta $ for some $\beta \in S$ , from the point of view of the ground model. The lemma only uses the parameters and (and $\tilde p$ , which is just ), so by absoluteness $M_\beta $ knows that the Lemma is forced by $p_{**}$ . Accordingly, we can find $P_\beta $ -names for U, $F_\xi $ , etc. in $M_\beta $ . Using the last item of Fact 5.55, we can strengthen $p_{**}$ to $p_{**}^2$ to canonically read these names:

Corollary 5.59. There is an $\alpha \in \Delta $ , a $p_{**}^2\le p_{**}$ in $P_\alpha $ and $P_{\alpha }$ -names for $:$ A condition $r_0\le \tilde p$ , a set U, and a sequence $(F_\xi , z^0_\xi , z^1_\xi ,\ell ^0_\xi , \ell ^1_\xi )_{\xi \in U}$ , such that the following holds for all $\beta \in S$ , where we set

$$\begin{align*}p^+_\beta\text{ to be the condition }p^2_{**} \wedge p_\beta\text{ where we strengthen } p_\beta(\beta)\text{ to } r_0. \end{align*}$$

(1) $\alpha $ , the condition $p^2_{**}$ and all the names are in $M_\beta $ .
(2) $p^2_{**}\Vdash U\subseteq C^{r_0}\subseteq \lambda $ club.
(3) For $k=0,1$ : .
(4) , where we define
$$\begin{align*}A_\xi\text{ to be }I^*({\ge}\xi,{<}\nu)\text{ with } \nu \text{ the } U\text{-successor of } \xi. \end{align*}$$

5.12. Finding the generator

In this section we use these $p^2_{**}$ , $r_0$ , $(F_\xi , z^0_\xi , z^1_\xi ,\ell ^0_\xi , \ell ^1_\xi )_{\xi \in U}$ .

We start working in $V_{\beta }=V[G_\beta ]$ , where we assume $p^2_{**}\in G_\beta $ .

Let and $\nu $ its U-successor. Set

(5.60)

For $F_\xi $ it is enough to use as input (the part in $A_\xi \setminus A^?_\xi $ is determined anyway by $r_0$ ), and every element of $2^{A^?_\xi }$ is compatible with $r_0$ (and thus a possible input for $F_\xi $ ). Identifying $2^B$ and ${\mathcal {P}}(B)$ as usual, we get

$$\begin{align*}F_\xi: {\mathcal{P}}(A^?_\xi)\to {\mathcal{P}}(A_\xi) \end{align*}$$

is such that $p^+_\beta /G_\beta $ forces

We now define

So in particular $p^+_\beta /G_\beta $ forces that

(5.61)

Note that for every (in $V_\beta $ that is) there is an $r'\le r_0$ forcing that . ( is club, so it is enough to leave freedom at $C'$ and we may assign arbitrary values everywhere else.)

Back in the ground model V, using the last item of Fact 5.55 again, we can strengthen $p^2_{**}$ to $p^3_{**}$ so thatFootnote ²⁰

(5.62)

$$ \begin{align} p^3_{**} \text{ canonically reads each of the following (countably many) sets:} \end{align} $$

• , , $r_0$ , , (actually, these are already read by $r^2_{**}$ ).
• The closure of these sets under , , finite unions, and finite intersections.

In particular, the (names for) all these sets are independent of $\beta \in S$ , modulo $p^3_{**}$ .Footnote ²¹

Lemma 5.63. (In V) .

Proof Let $q\le p^3_{**}$ in $P_\beta $ be arbitrary. We have to show that q does not force (in $P_\beta $ ) .

For and $k=0,1$ , use $r_0$ , $p_\beta ^+$ , $z^k_\xi $ and $\ell _\xi $ as in Corollary 5.59 and set $b^k_\xi := z^k_\xi \cap A_\xi ^?$ .

For $k=0,1$ , set . Note that $F(B^1)\setminus F(B^0)$ contains , a set of size $\lambda $ .

Pick increasing $(\beta _i)_{i< 4}$ in S with $\beta _0=\beta $ . Set $s:=q\wedge \bigwedge _{i<4} p^+_{\beta _i}\in P$ .

Now for each $i<4$ , strengthen $s(\beta _i)$ (i.e., $r_0$ ) as follows: At the even intervals in some way that together they honor majority; and at the odd intervals (where we do not have to leave freedom) to the value $B^{\operatorname {\mathrm {sgn}}(i)}$ (where $\operatorname {\mathrm {sgn}}(k)=0$ for $k=0$ and $1$ for $k=1,2,3$ ).

Accordingly, we have

or, when we split into the parts in and out of :

Now assume towards a contradiction that . Then we get

(5.64)

But on the other hand we have

Applying (5.64) to both sides of the last line, we get $F(B^0)=^*\operatorname {\mathrm {major}}_{i=1,2,3}F(B^{\operatorname {\mathrm {sgn}}(i)}) =F(B^1)$ , a contradiction.

Set

(5.65)

By choice of $p^3_{**}$ , and are canonically read by $p^3_{**}$ (and independent of $\beta $ ).

We now show that for . Again, here we are talking about $z\in V_\beta $ . To make that more explicit, let us formulate in the ground model V:

Lemma 5.66. For $\beta \in S$ ,

(Note that, other than $F(z)$ , will generally not be in $V_\beta $ , and we have to force with $p^{+}_\beta /G_\beta $ .)

Proof Work in $V_\beta $ . follows from Lemma 5.63, as .

Set . So by (5.61), $p^+_\beta /G_\beta \le r_0$ forces: . As , we get . Then (or equivalently, ) implies and thus . To summarize:

(*)

Now back in V assume towards a contradiction that some $q\le p^+_\beta $ forces that the lemma fails, i.e., that in $V_\beta $ is a counterexample (in the final extension). By absoluteness, we can assume that q and are in $M_\beta $ , in particular is a $P_\beta $ -name in $M_\beta $ . Strengthen $q\restriction \beta $ to canonically read . So for every $\beta '\in S$ , will be evaluated in $V_{\beta '}$ to the same $z\subseteq \lambda $ as in $V_\beta $ .

Chose a $\beta '$ above $\operatorname {\mathrm {supp}}(q)$ . Then we can strengthen $q\wedge p_{\beta '}$ at index $\beta '$ , i.e., $r_0$ , to some $r_1$ that forces . (Recall that we can fix the values in the odd intervals, as the even intervals still form a club). Let G be P-generic containing $q\wedge p^+_{\beta '}\wedge r_1$ . Then we have:

• The evaluation of in $V_{\beta '}$ , is the same as the evaluation of in $V_\beta $ , call it z.
• Also the evaluation of and F are the same $\beta $ and $\beta '$ (cf. (5.62)).
• is a counterexample (as this is forced by q).

In particular, and in the final extension.
• $p_{\beta '}\wedge r_1$ forces in $V_{\beta '+1}$ that ; also we have just seen that ; and so by (*), a contradiction.

For

, we define the following $P_\beta $ -names (independent of $\beta $ ):Footnote ²²

as well as

So the $p^3_{**}$ forces that for all $z \in V_\beta $ the following is forced by $p^{+}_\beta /G_\beta $ :

(5.67)

Lemma 5.68. $p^3_{**}$ forces: For almost all , $F^{\prime }_\xi $ is a Boolean algebra isomorphism from to .

Proof All and nothing: We claim that for almost all $\zeta $ , . Assume that . Then , and $\ell $ is not in , so there cannot be many such $\ell $ . Similarly $F^{\prime }_\zeta (\emptyset )=\emptyset $ for almost all $\zeta $ .

Unions: We claim that for almost all $\zeta $ , $F^{\prime }_\zeta (a)\cup F^{\prime }_\zeta (b)= F^{\prime }_\zeta (a\cup b)$ for all subsets $a,b$ of . Let $A\subseteq \lambda $ be the set of counterexamples, i.e., for $\xi \in A$ there are , and $a_\xi $ , $b_\xi $ subsets of such that $\ell _\xi \in \big (F^{\prime }_\xi (a_\xi )\cup F^{\prime }_\xi (b_\xi )\big )\Delta F^{\prime }_\xi (a_\xi \cup b_\xi )$ . Set $x:=\bigcup _{\xi \in A}a_\xi $ and $y:=\bigcup _{\xi \in A}b_\xi $ . Then $\ell _\xi $ is in $\big (F'(x)\cup F'(y)\big ) \Delta F'(x\cup y)=^*\emptyset $ , so A cannot be large.

Complements: We claim that for almost all $\xi $ , . Let A be the set of counterexamples, i.e., for $\xi \in A$ there is an and such that . Then $\ell _\xi $ is in , so A cannot be large.

Injectivity: We already know that union and complements (and thus disjointness) are preserved, so it is enough to show that a nonempty set is mapped to a nonempty set.

Assume this fails often, then we get an of size $\lambda $ such that , a contradiction.

Surjectivity: Assume surjectivity fails often; i.e., there are many not in the range of $F^{\prime }_\zeta $ . Let y be the union of those $b_\zeta $ . Pick $x\subseteq \lambda $ such that . So we can assume and so $F'(x)=^*y$ , which implies that for almost all $\zeta $ , a contradiction.

Lemma 5.69. For each $\beta \in S$ : $p^3_{**}$ forces (in $P_\beta $ ): There is a bijective such that for all (in $V_\beta $ ), $p^{+}_\beta /G_\beta $ forces .

Proof Every Boolean algebra isomorphism from $P(A)$ to $P(B)$ is generated by a bijection from A to B (the restriction to the atoms). So there is a with such that $\zeta \in U'$ implies that $F^{\prime }_\zeta $ is generated by some bijection . So $F'$ is generated by $g:=\bigcup _{\zeta \in U'} g_\zeta $ ; and we can change g into a bijection from to by changing less than $\lambda $ many values.

We now strengthen $p^3_{**}$ to some q to continuously read $f_{\text {gen}}$ (independently of $\beta $ ), again using Fact 5.55.

So to summarize, we have the following (where we start with the $\Delta $ -system $(M_\beta ,p_\beta )_{\beta \in S}$ of Section 5.10):

Corollary 5.70. There is $\alpha \in \Delta $ , $q\in P_\alpha $ stronger than all $p_\beta \restriction \beta $ and canonically reading $r_0\le \tilde p$ , , $f_{\text {gen}}$ , and , such that the following holds for all $\beta \in S$ :

• $q\wedge p_\beta $ with the conditionFootnote ²³ at index $\beta $ strengthened to $r_0$ is a valid condition, called $p^{++}_\beta $ .
• $\alpha $ , $p^{++}_\beta $ and the names are in $M_\beta $ .
• q forces in $P_\beta $ : , is a bijection, and if is in $V_\beta $ , then .

5.13. Putting everything together

Corollary 5.71. (Assuming $\lambda $ is inaccessible and $2^\lambda =\lambda ^+.$ ) P forces that every automorphism of $P^\lambda _\lambda $ is somewhere trivial.

Proof Assume towards a contradiction that some $p_*$ forces that is a nowhere trivial automorphism represented by .

As described in Section 5.10 we find a $\Delta $ -system $(M_\beta ,p_\beta )_{\beta \in S}$ with $p_\beta \restriction \beta \le p_*$ for all $\beta \in S$ , and we find q, , $f_{\text {gen}}$ as in Corollary 5.70, so in particular: $q\le p_\beta \restriction \beta $ for all S; and q forces that and that is a bijection.

As is nowhere trivial, $f_{\text {gen}}$ cannot be a generator, i.e., there is some with . Fix a name for this z and let $q^*\le q$ canonically read z.

Pick $\beta \in S$ above $\operatorname {\mathrm {dom}}(q^*)$ . So $q^*\wedge p^{++}_\beta $ is a valid condition, which forces that in the final extension $V[G]$ the following holds:

• with , as this is forced by $q^*$ .
• $z\in V_\beta $ , as $q^*$ canonically reads z.
• By Corollary 5.70 and as $p^{++}_\beta {\kern-1.5pt}\in{\kern-1.5pt} G$ , we get , a contradiction.

Acknowledgments

We thank an anonymous referee for numerous corrections. A variant of Sections 3 and 4, were included in the third author’s thesis.

Funding

Supported by Austrian Science Fund (FWF): grants [10.55776/P33420, 10.55776/P33895] (first author) and [10.55776/T1081] (third author); and Israel Science Foundation (ISF) grant 2320/23 (second author). For open access purposes, the authors have applied a CC BY public copyright license to any author accepted manuscript version arising from this submission.

Footnotes

This is publication number 1224 of the second author.

¹ Rudin [Reference Rudin7, Reference Rudin8] showed in the 1950s that CH implies that there is a non-trivial automorphism; Shelah [Reference Shelah9] showed that consistently all automorphisms are trivial. Further results can be found, e.g., in [Reference Dow1–Reference Farah and Shelah4, Reference Shelah and Steprāns11, Reference Shelah and Steprāns12, Reference Veličković15].

² See, e.g., [Reference Shelah and Steprāns13–Reference Veličković15].

³ A corrected version has been submitted (see https://shelah.logic.at/papers/990a/). This version again establishes the result only assuming inaccessibility.

⁴ That is, if $|I|<\kappa $ then $\bigvee _{i\in I}[A_i]=[\bigcup _{i\in I} A_i].$

⁵ A bijection $f:\omega \to \omega $ has infinitely many n such that $f(n)\ne n+1$ , and therefore an infinite set A such that $f"A$ is disjoint to $\{n+1:\, n\in A\}$ .

⁶ For $\delta =\lambda $ , it is enough that the $\xi _i$ converge to $\lambda $ . For $\delta <\lambda $ , we use that the $\xi _i$ are increasing and that $\sup (\xi _i)\ge \operatorname {\mathrm {cf}}(\delta )$ .

⁷ By which we mean $x\subseteq \eta ^r$ .

⁸ For example: For a p-condition Q, let be the set of odd elements of $C^p$ (or any other unbounded subset X of $C^p$ such that $C^p\setminus X$ is still club), and set . Note that for any , $\eta ^p\cup x$ defines a condition in Q (stronger than p). So if we fix any $p(0)\in P_1$ , and define the $P_1$ -name to be 0 iff is eventually constant to $0$ , then cannot be $(\{0\},\zeta )$ -decided by $p(0)$ for any $\zeta $ . And if $p(1)$ is any condition with , then is $(\{1\},1)$ -decided by $q:=(p(0),p(1))$ .

⁹ An example: $\operatorname {\mathrm {dom}}(p)=\operatorname {\mathrm {dom}}(q)=w=\{0,1\}$ , $\min (C^{p(0)})=\min (C^{q(0)})=\xi $ , and both $p(0)$ and $q(0)$ have trunk $a\in \operatorname {\mathrm {POSS}}^{Q}(\xi )$ . $p(0)$ forces that $p(1)=q(1)$ , that $\min (C^{p(1)})=\xi $ and that the trunk of $p(1)$ is either b or c (elements of $\operatorname {\mathrm {POSS}}^{Q}(\xi )$ ); both are possible with $p(0)$ . Now $q(0)\le ^+_\xi p(0)$ decides that the trunk of $p(1)$ is b. Then $q\le ^+_{w,\xi }p$ , and $(a,c)$ is in $\operatorname {\mathrm {poss}}(p,w,\xi )\setminus \operatorname {\mathrm {poss}}(q,w,\xi )$ . In particular $(a,c)\in \operatorname {\mathrm {poss}}(p,w,\xi )$ but p does not force that implies $c\in \operatorname {\mathrm {poss}}(p(1),\xi )$ .

¹⁰ Depending on the formal definition, we could/should add “modulo equivalence,” i.e., there is a ${\le }|\alpha |^\lambda $ -sized set Z of such pairs such that whenever p canonically reads in $P_\alpha $ then there is a such that and .

¹¹ To a condition that will generally not be coherent.

¹² Here we write $\beta _0$ instead of $\beta $ to stress the interaction with $\beta _1,\dots ,\beta _3$ , but recall that $\beta :=\beta _0$ .

¹³ As usual, $V_{\beta +1}$ denoted the $P_{\beta +1}$ -extension.

¹⁴ That is, $b_0=1-\operatorname {\mathrm {major}}_{i=1,2,3}(b_i)$ .

¹⁵ So $p_\beta \restriction \beta +1$ reads , but generally the whole $p_\beta $ may be required to force .

¹⁶ This means: $p^{\prime }_{**}\in M_\gamma $ for all $\gamma \in S$ , and there is a way (independent of $\gamma \in S$ ) to continuously read modulo $p^{\prime }_{**}$ from the generics below $\alpha $ , and for all $\gamma \in S$ we have that $p^{\prime }_{**}\wedge p_\gamma $ forces , and , where .

¹⁷ Or equivalently, a $(\zeta _0,\zeta _2)$ -candidate.

¹⁸ That is, if $(u_\alpha )_{\alpha <\lambda }$ is the canonical enumeration of U, then $\zeta \in U$ is in if $\zeta =u_{\delta +2n+1}$ for $\delta $ a limit (or 0) and $n\in \omega $ .

¹⁹ Assume y and x in $\operatorname {\mathrm {poss}}(r_0,\nu )$ are the identical restricted to $A^U_\xi $ . Then y defines the same $(x_i)_{i=1,2,3}$ and thus the same $F_\xi $ .

²⁰ We can do this for $\lambda $ many sets, of course; but we cannot assume, e.g., that for all $z\in V_\beta $ , let alone that each such is canonically read by $p^3_{**}$ .

²¹ But we need $p^{+}_\beta $ to force that these names have anything to do with .

²² More concretely, canonically read by $p^3_{**}$ , see (5.62).

²³ which is $p_\beta (\beta )=\tilde p$

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Article contents

ON AUTOMORPHISMS OF ${\mathcal {P}}(\lambda )/[\lambda ]^{<\lambda }$

Abstract

Keywords

MSC classification

1. Introduction

Contents

2. Definitions

Lemma 2.4 ( $\kappa>\aleph _0$ ).

Definition 2.5 ( $\lambda>\aleph _0$ regular.).

Fact 2.6 ( $\lambda>\aleph _0$ regular.).

Lemma 2.7 ( $\lambda>\aleph _0$ regular.).

3. Under MA, every automorphism is trivial for $\omega _1\le \lambda <2^{\aleph _0}$

Proof of the lemma

Proof of the theorem

4. For measurables, GCH implies a nontrivial automorphism

5. For inaccessible $\lambda $ , all automorphisms can be densely trivial

5.1. The single forcing $\boldsymbol Q$

5.2. Properness of $\boldsymbol Q$ : Fusion and pure decision

5.3. The iteration $\boldsymbol P$

5.4. Continuous reading and properness of $\boldsymbol P$

5.5. Canonical conditions

5.6. $\mathbf \Delta $ systems

5.7. Preservation of cofinalities, catching canonical names

5.8. Majority decisions

5.9. is in the $\beta +1$ -extension

Proof of the lemma

5.10. Fixing the $\mathbf \Delta $ -system

5.11. Local reading

5.12. Finding the generator

5.13. Putting everything together

Acknowledgments

Funding

Footnotes

References

REFERENCES

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