# NotesonForcingAxioms

NOTES ON FORCING AXIOMS 9013_9789814571579_tp.indd 114/11/13 10:21 AM LECTURE NOTES SERIES Institute for Mathematical Sciences, National University of Singapore Series Editors:Chitat Chong and Wing Keung To Institute for Mathematical Sciences National University of Singapore Published Vol. 16Mathematical Understanding of Infectious Disease Dynamics edited by Stefan Ma edited by Chitat Chong (National University of Singapore, Singapore), Qi Feng (National University of Singapore, Singapore), Yue Yang (National University of Singapore, Singapore), Theodore A. Slaman (University of California, Berkeley, USA), volume 26) Includes bibliographical references and index. ISBN 978-9814571579 (hardcover : alk. paper) 1. Forcing (Model theory) 2. Axioms. 3. Baire classes. I. Chong, C.-T. (Chi-Tat), 1949– editor. II. Feng, Qi, 1955– editor. III. Yang, Yue, 1964– editor. IV. Slaman, T. A. (Theodore Allen), 1954– editor. V. Woodin, W. H. (W. Hugh), editor. VI. Title. QA9.7.T63 2014 511.3--dc23 2013042520 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2014 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any or by any means, electronic or mechanical, including photocopying, recording or any ination storage and retri system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Printed in Singapore Contents Foreword by Series Editorsix Foreword by Volume Editorsxi Prefacexiii 1Baire Category Theorem and the Baire Category Numbers1 1.1The Baire category – a classical example . . . . . .1 1.2Baire category numbers. . . . . . . . . . . . . . . . . . . .3 1.3P-clubs. . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 1.4Baire category numbers of posets . . . . . . . . . . . . . . .6 1.5Proper and semi-proper posets. . . . . . . . . . . . . . . .8 2Coding Sets by the Real Numbers13 2.1Almost-disjoint coding . . . . . . . . . . . . . . . . . . . . .13 2.2Coding families of unordered pairs of ordinals . . . . . . . .15 2.3Coding sets of ordered pairs . . . . . . . . . . . . . . . . . .19 2.4Strong coding . . . . . . . . . . . . . . . . . . . . . . . . . .23 2.5Solovay’ s lemma and its corollaries . . . . . . . . . . . . . .31 3Consequences in Descriptive Set Theory41 3.1Borel isomorphisms between Polish spaces . . . . . . . . . .41 3.2Analytic and co-analytic sets. . . . . . . . . . . . . . . . .42 3.3Analytic and co-analytic sets under p ω1. . . . . . . . .43 4Consequences in Measure Theory45 4.1Measure spaces . . . . . . . . . . . . . . . . . . . . . . . . .45 4.2More on measure spaces . . . . . . . . . . . . . . . . . . . .48 5Variations on the Souslin Hypothesis51 5.1The countable chain condition . . . . . . . . . . . . . . . . .51 5.2The Souslin Hypothesis. . . . . . . . . . . . . . . . . . . .53 v viCONTENTS 5.3 A selective ultrafi lter from m ω1. . . . . . . . . . . . . .54 5.4The countable chain condition versus the separability . . . .56 6The S-spaces and the L-spaces61 6.1Hereditarily separable and hereditarily Lindel¨ of spaces . . .61 6.2Countable tightness and the S- and L-space problems. . .64 7The Side-condition 73 7.1Elementary submodels as side conditions . . . . . . . . . . .73 7.2Open graph axiom . . . . . . . . . . . . . . . . . . . . . . .75 8Ideal Dichotomies81 8.1Small ideal dichotomy . . . . . . . . . . . . . . . . . . . . .81 8.2Sparse set-mapping principle. . . . . . . . . . . . . . . . .85 8.3P-ideal dichotomy. . . . . . . . . . . . . . . . . . . . . . .88 9Coherent and Lipschitz Trees91 9.1The Lipschitz condition. . . . . . . . . . . . . . . . . . . .91 9.2Filters and trees. . . . . . . . . . . . . . . . . . . . . . . .94 9.3 Model rejecting a fi nite set of nodes. . . . . . . . . . . . .96 9.4Coloring axiom for coherent trees . . . . . . . . . . . . . . .98 10 Applications to the S-space Problem and the von Neumann Problem103 10.1 The S-space problem and its relatives. . . . . . . . . . . .103 10.2 The P-ideal dichotomy and a problem of von Neumann. .106 11 Biorthogonal Systems113 11.1 The quotient problem. . . . . . . . . . . . . . . . . . . . .113 11.2 A topological property of the dual ball . . . . . . . . . . . .121 11.3 A problem of Rolewicz . . . . . . . . . . . . . . . . . . . . .126 11.4 Function spaces . . . . . . . . . . . . . . . . . . . . . . . . .127 12 Structure of Compact Spaces133 12.1 Covergence in topology . . . . . . . . . . . . . . . . . . . . .133 12.2 Ultrapowers versus reduced powers . . . . . . . . . . . . . .137 12.3 Automatic continuity in Banach algebras. . . . . . . . . .143 13 Ramsey Theory on Ordinals147 13.1 The arrow notation . . . . . . . . . . . . . . . . . . . . . . .147 13.2 ω2? → (ω2,ω + 2)2. . . . . . . . . . . . . . . . . . . . . . . .148 13.3 ω1→ (ω1,α)2. . . . . . . . . . . . . . . . . . . . . . . . . .159 CONTENTSvii 14 Five Cofi nal Types169 14.1 Tukey reductions and cofi nal equivalence . . . . . . . . . . .169 14.2 Directed posets of cardinality at most ℵ1. . . . . . . . . . .170 14.3 Directed sets of cardinality continuum . . . . . . . . . . . .174 15 Five Linear Orderings177 15.1 Basis problem for uncountable linear orderings. . . . . . .177 15.2 Separable linear orderings . . . . . . . . . . . . . . . . . . .177 15.3 Ordered coherent trees . . . . . . . . . . . . . . . . . . . . .181 15.4 Aronszajn orderings. . . . . . . . . . . . . . . . . . . . . .186 16 Cardinal Arithmetic and mm189 16.1 mm and the continuum . . . . . . . . . . . . . . . . . . . . .189 16.2 mm and cardinal arithmetic above the continuum . . . . . .192 17 Refl ection Principles193 17.1 Strong refl ection of stationary sets. . . . . . . . . . . . . .193 17.2 Weak refl ection of stationary sets . . . . . . . . . . . . . . .195 17.3 Open stationary set-mapping refl ection . . . . . . . . . . . .197 Appendix A Basic Notions199 A.1 Set theoretic notions . . . . . . . . . . . . . . . . . . . . . .199 A.2 Δ-systems and free sets. . . . . . . . . . . . . . . . . . . .200 A.3 Topological notions . . . . . . . . . . . . . . . . . . . . . . .201 A.4 Boolean algebras . . . . . . . . . . . . . . . . . . . . . . . .202 Appendix B Preserving Stationary Sets205 B.1Stationary sets . . . . . . . . . . . . . . . . . . . . . . . . .205 B.2Partial orders, Boolean algebras and topological spaces . . .206 B.3A topological translation of stationary set preserving . . . .210 Appendix C Historical and Other Comments215 Bibliography217 This page intentionally left blankThis page intentionally left blank Foreword by Series Editors The Institute for Mathematical Sciences (IMS) at the National Univer- sity of Singapore was established on 1 July 2000. Its mission is to foster mathematical research, both fundamental and multidisciplinary, particu- larly research that links mathematics to other eff orts of human endeavor, and to nurture the growth of mathematical talent and expertise in research scientists, as well as to serve as a plat for research interaction between scientists in Singapore and the international scientifi c community. The Institute organizes thematic programs of longer duration and math- ematical activities including workshops and public lectures. The program or workshop themes are selected from among areas at the forefront of cur- rent research in the mathematical sciences and their applications. Each volume of the IMS Lecture Notes Series is a compendium of papers based on lectures or tutorials delivered at a program/workshop. It brings to the international research community original results or expository articles on a subject of current interest. These volumes also serve as a record of activities that took place at the IMS. We hope that through the regular publication of these Lecture Notes the Institute will achieve, in part, its objective of reaching out to the community of scholars in the promotion of research in the mathematical sciences. September 2013Chitat Chong Wing Keung To Series Editors ix This page intentionally left blankThis page intentionally left blank Foreword by Volume Editors The series of Asian Initiative for Infi nity (AII) Graduate Logic Summer School was held annually from 2010 to 2012.The lecturers were Moti Gitik, Denis Hirschfeldt and Menachem Magidor in 2010, Richard Shore, Theodore A. Slaman, John Steel, and W. Hugh Woodin in 2011, and Ronald Jensen, Gerald E. Sacks and Stevo Todorcevic in 2012. In all, more than 150 graduate students from Asia, Europe and North America attended the summer schools. In addition, two postdoctoral fellows were appointed during each of the three summer schools. These three volumes of lecture notes serve as a record of the AII activities that took place during this period. The AII summer schools was funded by a grant from the John Tem- pleton Foundation and partially supported by the National University of Singapore. Their generosity is gratefully acknowledged. October 2013 Chitat Chong Qi Feng∗ Yue Yang National University of Singapore, Singapore Theodore A. Slaman W. Hugh Woodin University of California at Berkeley, USA Volume Editors ∗Current address: Chinese Academy of Sciences, China. xi This page intentionally left blankThis page intentionally left blank Preface Baire category as a tool for showing the existence of interesting mathematical objects is well established in mathematics. The set-theoretic technique of Forcing brings this to another level of sophistication and potential applicability. The purpose of the notes is to expose some of these. This set of notes was build over the last ten years and tested on courses I gave in Paris (Spring of 2003), Toronto (Spring of 2005 and Fall of 2012), and Singapore (Summer of 2012).1 I would like to thank the students who took the courses for the help in organizing this set of lecture notes. Stevo Todorcevic Spring, 2013 1The Singapore course was a part of the AII Graduate Summer School jointly orga- nized and funded by the John Templeton Foundation and the Institute for Mathematical Sciences of the National University of Singapore. I would like to thank these institutions for their support. xiii This page intentionally left blankThis page intentionally left blank Chapter 1 Baire Category Theorem and the Baire Category Numbers 1.1The Baire category – a classical example Recall the following: Theorem 1 (Baire Category Theorem). Given any compact Hausdorff space, or complete metric space X, the intersection of any countable family of dense open subsets of X is dense in X. In particular, such an intersection is always nonempty. Theorem 2 (K. Weierstrass). There is a continuous nowhere diff erentiable function on the closed unit interval [0,1]. Proof. (S. Banach). Let C denote the set of all continuous functions on [0,1]. Note that C is a Banach space under the uni norm ?f?∞= sup{| f(x)|: x ∈ [0,1]}. It can easily be shown that C is a separable complete metric space under the metric ρ(f,g) = ?f − g?∞= sup{| f(x) − g(x)|: x ∈ [0,1]}. For each n 1 defi ne Fn= ? f ∈ C:(∃x ∈ ? 0,1 − 1 n ? )(∀0 0 such that | x − y|1Fn ? = C, and thus there is an f ∈ C \ ? n1Fn. For each x ∈ [0,1), any M 0, and any ? 0 it can be shown that there is a 0 ℵ1is called Martin’s Maximum (MM). It is easy to see that the mapping K ? → m(K) is non-increasing, and so we have ℵ1≤ mm ≤ m ≤ p ≤ m(R) ≤ c. 1.3P-clubs In this section, we explain the class of posets whose category number will give us another ulation of the invariant mm introduced above. Central to this goal is the notion of a P-club. Defi nition. Given a partial order P, a P-club is a family ?C(p) : p ∈ P? of subsets of ω1satisfying the following conditions: 1. (P-monotonicity): p ≤ q implies C(p) ⊇ C(q), and 2. (P-closedness): for each γ ∈ ω1\ C(p) there is a β β. Then, obviously C(s) ∩ (β,δ] ? = ∅. Let t ∈ P be a common extension of r and s. By P-monotonicity it follows that C(t)∩(β,δ] ? = ∅. However, this contradicts our choice of r. We then immediately have that C(q) ∩ S ? = ∅. It then follows that P is stationary set preserving. Corollary 4. mm ≤ m. The proof of the above Theorem is useful for pedagogical reasons that will become apparent in a future section. Note, however, that the following statement, which immediately implies the above Lemma, is also true: Proposition. Let P is a given c.c.c.-poset, and let ?C(p) : p ∈ P? be a P-club. Then the set {p ∈ P : C(p) is a club in ω1} is dense in P. Theorem 5. mm ≤ ω2. Proof. Consider the set P of all countable partial functions ω1→ ω2, and order P by inverse-inclusion. Note that P is σ-closed, as any countable 8CHAPTER 1. BAIRE CATEGORY union of countable sets is again countable. A fact to be proved in the next section shows that all σ-closed partial orders are stationary set preserving. For each α β. Then we trivially have that C(pn+1) ∩ (β,δ] ? = ∅. As r ≤ q ≤ pn+1, by P-monotonicity, this clearly contradicts the choice of r. As δ ∈ S, it immediately follows that {p ∈ P : C(p) ∩ S ? = ∅} is dense in P, and so P is stationary set preserving. Note that in the proofs that every partial order with the c.c.c. is sta- tionary set preserving, and that every σ-closed partial order is stationary set preserving, we have the following general scheme: Step 1: Start with a partial order P, a P-club ?C(p) : p ∈ P?, a stationary set S ⊆ ω1and a given condition p. Step 2: Take a countable elementary submodel M ≺ ?Hθ,∈?for some large enough θ which contains each of the objects from Step 1. Step 3: Extend the given condition p to some q. We then assume that q does not have some good property (in our cases, we assume that δ = ω1∩ M is not in C(q)). This leads us to a further extension r of q, each further extension of which is “bad”. Step 4: We defi ne some subset X of P in M which contains r, and which has some “nice” property of r. Step 5: Show that r is “refl ected” in X ∩M; i.e., show that r is compatible with some element s of X ∩ M. Step 6: Taking a common extension t of r and s, arrive at a contradiction. The most important step in the above is Step 5. Upon further inspec- tion, we may see that the actual defi nition of the set X was unimportant. All that was required of X was that it was a nonempty subset of P in M containing the condition r. Once this was accomplished, we are guaranteed that r will “refl ect” in X ∩ M. It is this notion of “refl ecting” that will be extremely important to the concept of properness. 10CHAPTER 1. BAIRE CATEGORY Defi nition. Let P be a partial order, and M a countable elementary sub- model of ?Hθ,∈? for some large enough θ with P ∈ M. We say that q ∈ P is (M,P)-generic (or, simply M-generic) if for every r ≤ q and X ⊆ P such that X ∈ M and r ∈ X there is ¯ r ∈ X ∩ M such that r and ¯ r are compatible. The notion of properness will simply state that there are always many (M,P)-generic conditions: Defi nition. A partial order P is called proper if for each countable ele- mentary submodel M ≺ ?Hθ,∈,? for large enough θ the set and for every p ∈ P ∩ M there is q ≤ p such that q is (M,P)-generic. The proofs given above readily give us the following: Corollary 7.1. Every c.c.c. partial order is proper. 2. Every σ-closed partial order in proper. Corollary 8. Every proper partial order is stationary set preserving. rcise. Give an example of a proper poset P that is neither c.c.c. nor is σ-closed. Can you fi nd such a poset P to be of cardinality ℵ1or ℵ2? rcise. Let T be a tree of height ω1.Under which conditions is T as a forcing notion proper? There is another less restrictive condition on a poset P that is still stronger than the condition of preserving all stationary subsets of ω1. Defi nition. Let P be a partial order, and M a countable elementary sub- model of ?Hθ,∈? for some large enough θ with P ∈ M. We say that q ∈ P is (M,P)-semi-generic (or, simply M-semi-generic) if for every r ≤ q and every partial function f : P → ω1such that f ∈ M and r ∈ dom(f) there exist α ∈ M ∩ ω1and ¯ r ∈ f−1(α) such that r and ¯ r are compatible. Defi nition. A partial order P is called semi-proper if for each countable elementary submodel M ≺ ?Hθ,∈,?for large enough θ the set and for every p ∈ P ∩ M there is q ≤ p such that q is (M,P)-semi-generic. Clearly, every proper poset is semi-proper. rcise. Give an example of a semi-proper poset that is not proper. What is the minimal cardinal κ for which you can fi nd a semi-proper poset that does not preserve stationary subsets of [κ]ℵ0? rcise. Let T be a tree of height ω1.Under which conditions is T as a forcing notion semi-proper? 1.5. PROPER AND SEMI-PROPER POSETS11 Theorem 9. Every semi-proper posets preserves all stationary subsets of ω1. Proof. Let (Cp: p ∈ P) be a given P-club, Let E be a given stationary subset of ω1 and let ¯ p be a given condition of P.WE need to fi nd q ≤ ¯ p such that C(q) ∩ E ? = ∅.Choose a countable elementary submodel M of some large enough structure of the (Hθ,∈) such that δ = M ∩ ω1 belongs to E and such that M contains all the relevant objects such as P, (Cp: p ∈ P),and E.Choose p ≤ ¯ p such that p is (M,P)-semi-generic and then choose q ≤ p such that C(q) \ δ ? = ∅. Claim 9.1. δ ∈ C(q). Proof. Otherwise, we can fi nd r ≤ q and γ ℵα. Chapter 2 Coding Sets by the Real Numbers 2.1Almost-disjoint coding Defi nition. A family A of infi nite subsets of N is called almost disjoint (a.d.) if A ∩ B ∈ Fin for any distinct A,B ∈ A. Proposition (W. Sierpinski, 1930’ s). There is an a.d. family A of size c. Proof. We will instead construct an almost disjoint family of infi nite subsets of Q. This is clearly suffi cient, and any bijection Q → N will translate such a family into an a.d. family of infi nite subsets of N. For each real number x fi x a nonrepeating sequence {q(x) n}n 0} : m ∈ ω \ {n},{m,n} / ∈ S ? . It is easy to check that {An}n max(Xq) and also ? / ∈ ? β∈FqAβ. Claim 11.3. The following subsets of P are dense-open in ?P,≤?: (i) For each α k, so p ∈ D(2) α,k. By Claim 11.2 we also have p ≤ q. Suppose that p ≤ q ∈ P and q ∈ D(2) α,k. As Xp ⊇ Xq, we clearly have (Xp∩ Aα) \ {0,.,k} ⊇ (Xq∩ Aα) \ {0,.,k} ? = ∅. Thus p ∈ D(2) α,k. (iii) Let Γ ⊆ γ be fi nite, and let k ∈ N. Suppose that q ∈ P but q / ∈ D(3) Γ,k. By (3)γwe may choose some ?∈ N\?β∈Γ∪FqAβwith ? k,max(Xq). Defi ne p = ?Xq∪{?},Fq?. By choice we have ?∈ Xp\?β∈ΓAβ. Thus, as ? k, we have p ∈ D(3) Γ,k. By Claim 11.2 we also have p ≤ q. Suppose that p ≤ q ∈ P and q ∈ D(3) Γ,k. As Xp ⊇ Xqwe trivially have (Xp\ ? β∈Γ Aβ) \ {0,.,k} ⊇ (Xq\ ? β∈Γ Aβ) \ {0,.,k} ? = ∅. Thus p ∈ D(3) Γ,k. (iv) Let α k.By (3)γwe may choose some ? ∈ N \ ? β∈FqAβ with ? ??,max(Xq ). Defi ne p = ?Xq∪ {?},Fq?. Clearly ??/ ∈ Xpand so by choice we have ??∈ Aα\ (Xp∪ ? β∈ΓAβ). 18CHAPTER 2. CODING SETS BY THE REAL NUMBERS Also by choice we have k ≤ ?? ω ω ω ω1 Corollary (Martin and Solovay). If p ω1, then the following are equivalent 1. There is an uncountable co-analytic set without a perfect subset. 2. Every set of size at most ℵ1in a complete separable metric space is co-analytic. 3. Every union of at most ℵ1Borel set is a continuous image of a co- analytic set. 44 CHAPTER 3. CONSEQUENCES IN DESCRIPTIVE SET THEORY Proof. (2) implies (1) is trivial, since every set of size ℵ1contains no perfect set (we are assuming p ω1). Now, we prove (1) implies (2). Let C be a co-analytic set without a perfect subset in a Polish space X. Then by Theorem 24, C is the union of ℵ1many Borel sets, i.e., C = ? α 0. Suppose we cannot fi nd such H0and H1, i.e., for any distinct H0,H1∈ H, μ(H0∩ H1) = 0. Since Q is dense and countable in R and H is uncountable, there is ? 0 and an uncountable K ⊆ H such that μ(H) ? for any H∈ K. By fi niteness of μ(K), let