Alternative Cichoń Diagrams and Forcing Axioms Compatible with CH
aa r X i v : . [ m a t h . L O ] A ug Alternative Cicho´n Diagrams and Forcing Axioms Compatible with CH by Corey Bacal Switzer
A dissertation submitted to the Graduate Faculty in Mathematics in partial fulfillment ofthe requirements for the degree of Doctor of Philosophy, The City University of New York.2020ic (cid:13)
Corey Bacal Switzer
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Alternative Cicho´n Diagrams and Forcing Axioms Compatible with CH by Corey Bacal Switzer
This manuscript has been read and accepted by the Graduate Faculty in Mathematics insatisfaction of the dissertation requirement for the degree of Doctor of Philosophy.
Professors Gunter Fuchs and Joel DavidHamkins
Date Co-Chairs of Examining Committee
Professor Ara Basmajian
Date Executive Officer
Professor Gunter FuchsProfessor Joel David HamkinsProfessor Arthur Apter
Supervisory Committee
The City University of New York vAbstract
Alternative Cicho´n Diagrams and Forcing Axioms Compatible with CH by Corey Bacal Switzer
Advisors: Professors Gunter Fuchs and Joel David HamkinsThis dissertation surveys several topics in the general areas of iterated forcing, infinite com-binatorics and set theory of the reals. There are four largely independent chapters, the firsttwo of which consider alternative versions of the Cicho´n diagram and the latter two considerforcing axioms compatible with CH . In the first chapter, I begin by introducing the notionof a reduction concept , generalizing various notions of reduction in the literature and showthat for each such reduction there is a Cicho´n diagram for effective cardinal characteristicsrelativized to that reduction. As an application I investigate in detail the Cicho´n diagramfor degrees of constructibility relative to a fixed inner model W | = ZFC .In the second chapter, I study the space of functions f : ω ω → ω ω and introduce 18 newhigher cardinal characteristics associated with this space. I prove that these can be organizedinto two diagrams of 6 and 12 cardinals respecitvely analogous to the Cicho´n diagram on ω .I then investigate their relation to cardinal invariants on ω and introduce several new forcingnotions for proving consistent separations between the cardinals.The third chapter concerns Jensen’s subcomplete and subproper forcing. I generalizethese notions to the (seemingly) larger classes of ∞ -subcomplete and ∞ -subproper. I showthat both classes are (apparently) much more nicely behaved structurally than their non- ∞ -counterparts and iteration theorems are proved for both classes using Miyamoto’s niceiterations. Several preservation theorems are then presented. This includes the preservationof Souslin trees, the Sacks property, the Laver property, the property of being ω ω -boundingand the property of not adding branches to a given ω -tree along nice iterations of ∞ -subproper forcing notions. As an application of these methods I produce many new models ofthe subcomplete forcing axiom, proving that it is consistent with a wide variety of behaviorson the reals and at the level of ω .The final chapter contrasts the flexibility of SCFA with Shelah’s dee-complete forcing andits associated axiom
DCFA . Extending a well known result of Shelah, I show that if a tree ofheight ω with no branch can be embedded into an ω -tree, possibly with branches, then itcan be specialized without adding reals. As a consequence I show that DCFA implies thereare no Kurepa trees, even if CH fails. edication For Paki, who taught me to ask questionsandMahala, who showed me how to find answersvi cknowledgments
It’s hard for me to put into words how grateful I am to all of the people who helped meenormously in this undertaking. First and foremost I would like to thank my advisors Joeland Gunter. Thank you both so much.Gunter, you are an inspiring mentor. Thank you for all of our meetings at the coffeeshop, for all that you taught me, for your patience, and for teaching me to be more carefulin checking myself (someday I hope to get this right).Joel, your enthusiasm is infectious. Thank you for all you have shown me about how tothink about mathematics. It is something I am grateful to be able to carry forward. Thankyou also for my visit to Oxford, it was an amazing experience.Next I want to thank Roman Kossak. Roman, even though you weren’t my advisor I amso grateful for all of our time together and all that you taught me. Thank you for introducingme to models of PA and inviting me to give my first ever talk.Vika, thank you for allowing me to tag along and play organizer with you, for sharingwith me your perspective on math, and for inviting me to help organize the conference, evenif it didn’t happen.I would like to thank all of the amazing faculty at CUNY that I had the pleasure tointeract with during these last four years, particularly the logic group. Thanks to all of thelogicians that allowed me to bother you with questions continuously every Friday during andbetween the seminars. viiiiiThanks to Arthur Apter for agreeing to be on my defense committee and Alf Dolich foragreeing to be on my oral exam committee.Thanks to Alice Medvedev for all of her invaluable advice.At CUNY I want to also thank all of the staff that I interacted with over the years.I especially want to thank Debbie Silverman at the Graduate Center and Norma Moy atHunter.Next I would like to thank all the (non-CUNY) logicians and mathematicians who metwith me and patiently answered all my annoying questions. In particular thanks so muchto Ali Enayat, Andrew Brooke-Taylor, Asaf Karagila, Brent Cody, Chris Lambie-Hanson,Giorgio Venturi, Grigor Sargsyan, Henry Towsner, Hiroshi Sakai, J¨org Brendle, Sean Cox,Saka´e Fuchino, Simon Thomas, and Vera Fischer. Thanks especially to Boban Veliˇckovi´c forteaching me forcing, supervising my M2 thesis in Paris and all of the conversations we havehad since then.I want to thank Mirna Dˇzamonja for introducing me to set theory, a gift I couldn’t everhope to repay.Thanks to all of the great friends I have made throughout my PhD. Thanks to all of theset theory and MOPA students, Alex, Kameryn, Kaethe, Miha, Eoin, Ryan, Ben and Athar.Double thanks to Kameryn for pushing me to go to Brazil and traveling with me to Braziland Japan. Thank you to Iv´an and Micha l. Thanks to James for the music.Thanks to Sam, Oliver, May, Bo, Jesse and Alan.Thank you to Alfie the dog for not being dead yet and being so dumb.Thank you to my family. To my parents Jeff and Karen for all their love and supportand putting up with me living upstairs from them. To my sister Shauna for the chats, thetravels and being smarter than I could ever be. To my in-laws Adam, Sylvaine and Noah forincluding me in your family. To Ron, for teaching me what a shillelagh is, and not beatingme with one. To Daniel, Jay, Julia, Matt, Amanda, Alessandra, James, Ian, Brooke, Connor,xChase, Gail, Jeff, Ginny, Michael, Susan, Jesse and Becky. Merci `a la famille Franc`es-Combespour m’avoir accueillit `a bras ouverts en France quand j’ai commenc´e ces ´etudes. Thank youto my grandmother Annie for everything you do and did.This academic year we lost my grandfather, Joe “Paki” Bacal. Words cannot expresshow much he meant to me. Even though he never would have understood this thesis, hewould have read it anyway. I like to think he would have found in it shadows of what hetaught me.Thank you Mahala. You are my best friend and I love you. Everyday you amaze me,frustrate me and inspire me. Without you I never would have even gone to grad school yetalone finished it.I never expected to finish my PhD during a global pandemic. The events of the last fewmonths have been difficult and at times, overwhelming. More so than ever I feel so gratefulfor everything, especially to have had the opportunity to study something I love so much inan as amazing environment as CUNY. On that note I want to finish these acknowledgmentsby thanking all of the students I have had the privilege to teach at Hunter College. I didn’tgo to grad school to teach, but it ended up being an endless source of joy and frustration,especially when the pandemic raged.Thank you to you all, you’ve taught me so much. ontents
Contents xList of Figures xiii0 Introduction 1 ≤ W . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.3.1 Sacks Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.3.2 Cohen Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.3.3 Random Real Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.3.4 Laver Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.3.5 Rational Perfect Tree Forcing . . . . . . . . . . . . . . . . . . . . . . 261.3.6 Hechler Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.3.7 Eventually Different Forcing . . . . . . . . . . . . . . . . . . . . . . . 301.3.8 Localization Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.3.9 Cuts in the Diagram and the Analogy with Cardinal Characteristics . 37x ONTENTS xi1.4 Achieving a Full Separation in the ≤ W -Cicho´n Diagram and the axiom CD ( ≤ W ) 411.5 Open Questions and Further Work . . . . . . . . . . . . . . . . . . . . . . . 46 LOC -Forcing . . . . . . . . . . . . . . . . . . . . . . . . 722.4 Conclusion and Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 ∞ -Subversion Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.2 A Subpar Primer on Nice Iterations . . . . . . . . . . . . . . . . . . . . . . . 823.3 Nice Iterations of ∞ -Subversion Forcing . . . . . . . . . . . . . . . . . . . . 873.4 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043.5 Preservation Theorems for the Reals . . . . . . . . . . . . . . . . . . . . . . 1083.6 Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 <ω -Properness and Trees . . . . . . . 1194.1.1 Strengthening Properness . . . . . . . . . . . . . . . . . . . . . . . . 1194.1.2 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1224.2 Specializing a Wide Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.3 DCFA implies there are no Kurepa Trees . . . . . . . . . . . . . . . . . . . . 139
ONTENTS xii4.4 Looking Forward: Some Cardinal Characteristics and Many Open Questions 143
Bibliography 145 ist of Figures ≤ W Cicho´n Diagram. Each one can be achieved by aproper forcing over W . White means that the node is not empty while yellowmeans that it is. No distinction is made between different non-empty nodes.Note that the trivial cut where all nodes remain empty is not shown. . . . . 381.13 Full Separation of the ≤ W -diagram . . . . . . . . . . . . . . . . . . . . . . . 421.14 Integrating the Combinatorial Nodes and the Nodes for Measure and Category 49xiii IST OF FIGURES xiv2.1 Higher Dimensional Cardinal Characteristics Mod the Null Ideal . . . . . . . 522.2 Higher Dimensional Cardinal Characteristics Mod the Meager and σ -CompactIdeals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.3 The Easy Cases of the Higher Cicho´n Diagram . . . . . . . . . . . . . . . . . 55 hapter 0Introduction Paul Cohen’s discovery of forcing, [14, 15, 16], revolutionized set theory. The techniquenot only provided a flexible method for producing new models of
ZFC but also opened upuncountably many possibilities for consistent models of the real line, telling vastly differentstories about its topological and measure theoretic properties. Similarly, infinite combina-torics, especially concerning trees and their relatives, were soon seen to be equally malleable.These early results were extended by the discovery of iterated forcing, first seen in [19], andthen, in the context of forcing axioms in [46, 39].In this thesis I explore several topics roughly related to forcing constructions and thecontinuum. While each chapter is essentially independent, there are certain thematic threadsthat tie them together. Specifically, in each chapter I look at some aspect of set theory whichis usually studied in the context of the failure of CH , but I modify it to be compatible withthe continuum taking many values, including ℵ . Here is a brief outline of the structure ofthe thesis.In the first chapter I introduce the notion of a reduction concept , generalizing the idea ofTuring reduction and prove that for any “reasonable” reduction concept there is a correspond-ing Cicho´n diagram. This extends known results of Rupprecht [44] and others [12, 37, 27]1 HAPTER 0. INTRODUCTION W | = ZFC . I show that this diagram is complete in the sense that any two-valued cut in the diagram is consistent, however in most interesting cases the diagram splitsinto more than one set in contrast to the cardinal case. Most of the individual results hereare known, however collectively the new perspective sheds light on the relation between theforcing notions used in cardinal characteristics and their effective counterparts. The mainresult of this chapter is that there is a proper forcing P ∈ W so that every separation in thediagram can be made simultaneously in a way that is preserved by any further forcing. Thisleads to a new axiom, CD ( ≤ W ), which essentially states that the Cicho´n diagram for ≤ W is as complicated as possible. I show that CD ( ≤ W ) is compatible with CH , but a strongerversion implies all of the cardinals in the Cicho´n diagram are greater than ℵ . The resultsof this chapter appear in print in [48].In the second chapter I consider a different generalization of cardinal characteristics of thecontinuum. Much work recently has considered generalizations of well known characteristicsto the space κ κ . Here, I look instead at the space ( ω ω ) ω ω of functions from ω ω to ω ω .Eighteen new cardinal characteristics for this space are introduced and provable inequalitiesand consistent separations for these cardinals are investigated. I show that several diagramssimilar to the Cicho´n diagram exist for these spaces. I also show that various constellationsfor the cardinal characteristics on ω play a role in the values of these ones and I also introducethree new forcing notions for proving separations between the new cardinals. The results ofthis chapter appear in print in [47].In the third and fourth chapters I switch gears and turn my attention from cardinalcharacteristics to forcing axioms compatible with CH . The existence of such axioms has longbeen of interest in set theory see, for example, [4, 20, 3], however, Jensen’s recent work insubcomplete forcing ([33]) represents a breakthrough. The subcomplete forcing axiom is a HAPTER 0. INTRODUCTION CH and even ♦ . In the third chapter I investigate therole of the continuum in this axiom and show that many consequences of CH and ♦ can bepreserved while forcing CH to fail in a model of SCFA . This involves proving new iterationand preservation theorems for subcomplete and subproper forcing. The type of iteration Iuse is the nice support iteration of Miyamoto [41]. One of the unexpected advantages of thisapproach is it allows a (seemingly) more general class of forcing notions beyond subcompleteand subproper to be iterable. I dub these ∞ -subcomplete and ∞ -subproper forcing andconsider the structural properties of these classes as well. The work in this chapter alsoappears in print as part of the larger work [25].In the fourth chapter, to contrast my work on subcomplete forcing I look at the axiom DCFA , first considered alongside the assumptions of CH and 2 ℵ = ℵ in [45] and less re-strictively by Jensen in [32]. While DCFA is also compatible with CH , in contrast to SCFA this axiom seems to effect the universe at the level of the continuum and ω . I show thatit implies that there are no Kurepa trees, a result sketched by Shelah in [45]. There thestatement assumes the additional assumptions of CH and 2 ℵ = ℵ , though they are notused. To prove this theorem I generalize Shelah’s idea, introducing a forcing notion whichcan specialize certain wide Aronszajn trees of height ω and can be iterated without addingreals. I explore a few other applications of this forcing. The work in this chapter appears in[49].Since each chapter is essentially independent I provide preliminaries at the beginning ofeach chapter. In some cases, a definition is listed in two chapters for the convenience of thereader. However every effort has been made to uniformize notation. Also, the work in eachchapter has led to ongoing research and I briefly outline at the end of each chapter openquestions and current investigations along the lines of the content there. HAPTER 0. INTRODUCTION Let me end this introduction by fixing some notation and recalling some basic definitionsthat will be used in every chapter. Overall, most notation is standard, and all undefinedterms can be found in the well known monographs [38] and [31]. Also, I use the monograph[7] as the standard reference for cardinal characteristics of the continuum and occasionallyrefer to the survey article [10] as well. Throughout this thesis I use the convention that if P is a forcing notion and p, q ∈ P with q ≤ p then q is stronger than p . One notationalconvention which varies slightly from the norm is that for the most part I will let letters like x, y, z, ... stand for reals (elements of 2 ω , ω ω , etc) and letters like f, g, h, ... stand for functionsbetween uncountable Polish spaces. This will be relevant in particular in chapter 2 where Iwill frequently refer to both.Let I be a non-trivial ideal whose dual filter is non-principle. A set is I -positive if it’snot in I and is I -measure one if its complement is in I . For every such ideal I on a set X we naturally associate four cardinal characteristics.1. The additivity number : add ( I ), the least size of a family of sets in I whose union isnot in I .2. The uniformity number : non ( I ), the least size of an I -positive set.3. The covering number : cov ( I ), the least size of a family of sets in I needed to cover X .4. The cofinality number : cof ( I ), the least size of a family of sets in I which is cofinal in I with respect to inclusion.Given any set X and a relation R on X , we say that an element x ∈ X is an R - bound for a set A ⊆ X if for every a ∈ A we have that a R x . A set is R - bounded if it has an R -bound. It’s R - unbounded otherwise. A set D ⊆ X is R - dominating if for every y ∈ X there is a d ∈ D so that y R d . For any such X and R I write b ( R ) for the least size of an HAPTER 0. INTRODUCTION R -unbounded set and d ( R ) for the least size of an R -dominating set. If Q = ( Q, ≤ Q ) is apartially ordered set then I also write b ( Q ) and d ( Q ) for b ( ≤ Q ) and d ( ≤ Q ) respectively.I let µ denote the Lebesgue measure on ω ω (or any other oft-encountered Polish spaceunder consideration). The symbols N , M , K denote the null ideal, the meager ideal andthe ideal generated by σ -compact subsets of ω ω respectively. If x, y ∈ ω ω then x ≤ ∗ y if andonly if for all but finitely many n we have x ( n ) ≤ y ( n ) and b = b ( ≤ ∗ ), d = d ( ≤ ∗ ). Recallthat A ∈ K if and only if A is ≤ ∗ -bounded, see the proof of [10, Theorem 2.8]. The relevantproperties that all three of these ideals share is that they are non-trivial σ -ideals containingall countable subsets of ω ω and have a Borel base: every element of each ideal is covered bya Borel set in that ideal. In the case N and M the fact that the underlying set is ω ω , asopposed to any other perfect Polish space is unimportant in this thesis, however, it obviouslymatters for K since many Polish spaces are themselves σ -compact and hence K on such aspace is trivial.Implicit in several of these chapters is the classical Cicho´n diagram, see [7, Chapter 2].This diagram relates the cardinal characteristics for N , M and b and d (which are thecardinal invariants associated with K ). It is produced below for reference, note that x → y means that x is ZFC -provably less than or equal to y . add ( M ) non ( N ) cof ( M ) b d cov ( N ) add ( N ) cof ( N ) ℵ non ( M ) c cov ( M )Figure 1: The Cicho´n DiagramFinally we recall basic terminology of forcing axioms. If Γ is a definable class of forcingnotions and κ is a cardinal then Martin’s Axiom For
Γ, sometimes also called the forcingaxiom for
Γ, which is denoted MA κ (Γ), is the statement that for any P ∈ Γ, and any κ HAPTER 0. INTRODUCTION h D α | α < κ i there is a filter G ⊆ P so that for any α < κG ∩ D α = ∅ . If Γ is the class of c.c.c. forcing notions then MA denotes ∀ κ < ℵ MA κ (Γ)holds. If Γ is the class of proper forcing notions then we write PFA for MA ℵ (Γ). If Γ is theclass of stationary set preserving forcing notions then we write MM for MA ℵ (Γ). hapter 1The Cicho´n Diagram for Degrees ofRelative Constructibility In this chapter I introduce the notion of a reduction concept and tie it to cardinal character-istics. While the notion of a reduction concept is rarely written down explicitly in publishedwork (though see [37, 27]), it has been implicit in the literature since the beginning of the20th century. Turing reductions, polytime reductions, arithmetic reductions and degrees ofconstructibility are all examples of reduction concepts. Each comes with its own notion ofdegree. In this chapter I show that each such degree theory can formulate a variety of “high-ness properties” analogous to some common cardinal characteristics of of the continuum andthe implications between these highness properties resemble those of the standard Cicho´ndiagram for cardinals. The case of Turing degrees was worked out in [12], piggybacking offwork from [44], so my main contribution here is generalizing the result to the general case.Similar ideas have been explored in Section 5 of [27] and in [37] . In contrast with thosepapers though I work more on the level building analogues of the Cicho´n diagram than inconsidering relations between various types of reducibilities in higher computability theory Thanks to the anonymous referee of [48] for pointing this out to me. HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES W | = ZFC . In this case I show, amongstother things, that there is a proper forcing P ∈ W so that in W P the Cicho´n diagram for ≤ W is fully separated in the sense that every node is non-empty and all consistent non-implications are simultaneously realized. This set up is expressed as the axiom CD ( ≤ W ) andsome consequences of it are investigated. Before beginning in earnest I list a few definitions and facts which will be used throughoutthis chapter. The first definition will in fact be essential throughout this thesis.
Definition 1.1.1 (Combinatorial relations) . Let x and y be elements of ω ω . Then1. x ≤ ∗ y if and only if for all but finitely many k we have x ( k ) ≤ y ( k ). In this case wesay that y eventually dominates x .2. x = ∗ y if and only if for all but finitely many k we have x ( k ) = y ( k ). In this case saythat y is eventually different from x . Note that the negation of = ∗ is infinitely oftenequal , not eventual equality.3. Let z ∈ ω ω and recall that a z - slalom is a function s : ω → [ ω ] <ω such that for all n ∈ ω the set | s ( n ) | ≤ z ( n ). In the case where z is the identity function call s simply a slalom .I denote the space of all slaloms as S . This space is can be treated as homeomorphicto Baire space in the obvious way, see [43] for the details of a particularly useful codingof this correspondence. HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES s , I write x ∈ ∗ s if and only if for all but finitely many k we have x ( k ) ∈ s ( k ). In this case say that x is eventually captured by s .The cardinal characteristics associated with these relations have nice descriptions in termsof the ideals N , M and K . First, for ∈ ∗ there is a relation with N . Fact 1.1.2 (Bartoszy´nski, see Theorem 2.3.9 of [7]) . The following equalities are provablein
ZFC .1. b ( ∈ ∗ ) = add ( N ) d ( ∈ ∗ ) = cof ( N )Next, for = ∗ there is a relation with M . Fact 1.1.3 (Bartoszy´nski, see Thereoms 2.4.1 and 2.4.7 of [7]) . The following equalities areprovable in
ZFC .1. b ( = ∗ ) = non ( M ) d ( = ∗ ) = cov ( M )Finally, for ≤ ∗ , there is a relation with K . Fact 1.1.4 (See Theorem 2.8 of [10]) . The following equalities are provable in
ZFC .1. b = add ( K ) = non ( K ) d = cov ( K ) = cof ( K ) . When attempting to control these relations while forcing, the following three propertiesof forcing notions will be useful.
HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES Definition 1.1.5. ([7, Definition 6.3.37]) Let P be a forcing notion. We say that P has the Sacks property if given any p ∈ P and any name ˙ x , so that p (cid:13) ˙ x : ω → V is a functionthen there is a q ≤ p and a function y : ω → V in V such that for all n , q (cid:13) ˙ x (ˇ n ) ∈ ˇ y (ˇ n )and | y ( n ) | ≤ n . Slightly less formally this means that every new real (in fact ω sequence)is caught in an old slalom . In particular, all reals added by P can be captured by a slalomfrom the ground model. Definition 1.1.6. ([7, Definition 6.3.1]) Let P be a forcing notion. We say that P is ω ω - bounding if for each p ∈ P and each P -name ˙ x , if p (cid:13) ˙ x : ˇ ω → ˇ ω there is a y ∈ ω ω ∩ V and a q ≤ p so that q (cid:13) ˙ x ≤ ∗ ˇ y . In other words, every new real is ≤ ∗ -dominated by some old real.Note that this implies that the reals of V are dominating in V P . Definition 1.1.7. ([7, Definition 6.3.27] ) Let P be a forcing notion. We say that P has the Laver Property if given any p ∈ P and any name ˙ x , so that p (cid:13) ˙ x : ω → ω is a function whichis ≤ ∗ -bounded by a ground model real then there is a q ≤ p and a function y : ω → V in V such that for all n , q (cid:13) ˙ x (ˇ n ) ∈ ˇ y (ˇ n ) and | y ( n ) | ≤ n . In words, this says that every new realwhich is bounded by an old real is caught in an old slalom . Note that the Laver propertyplus ω ω -bounding is equivalent to the Sacks property.All three of these properties are preserved by countable support iterations of properforcing notions. See [7, Chapter 6]. Let us think of cardinal characteristics of the continuum in terms of small and large setsrelative to some relation giving this notion of smallness and largeness. For example, recallthat a family of reals A is ( ≤ ∗ ) - unbounded if for all x ∈ ω ω there is some y ∈ A such that In fact the function bounding n
7→ | y ( n ) | can be any function from V tending to ∞ , see [10, p. 86]. Again, n n can be replaced with any ground model function tending to infinity. HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES y (cid:2) ∗ x . The smallest cardinality of an unbounded family is called the bounding number ,denoted b = b ( ≤ ∗ ). Dually, a family of reals A ⊆ ω ω is ( ≤ ∗ ) - dominating if for all y ∈ ω ω there is a x ∈ A such that y ≤ ∗ x . The least size of a dominating family is called the dominating number , denoted d = d ( ≤ ∗ ). Intuitively one thinks of bounded families as being“small” and dominating families as being “big”. Thus, heuristically one might think of b asthe least size of a set that’s not “small” and d as the least size of a set that’s “big”. To obtainan analogy in the computable world, the authors of [12] define B ( ≤ ∗ ) as the set of oraclescomputing a function x such that y ≤ ∗ x for each computable function y and D ( ≤ ∗ ) as theset of oracles computing a function x such that x (cid:2) ∗ y for all computable y . In other words B ( ≤ ∗ ) is the set of oracles which can compute a witness to the fact that the computablefunctions are “small” and D ( ≤ ∗ ) is the set of oracles which can compute a witness to the factthat the computable functions are not “big”. Moreover, these sets turn out to correspondto “highness” properties of Turing degrees that are well studied in computability theory.Specifically, by a theorem of Martin (cf [12, pp. 3]), B ( ≤ ∗ ) is the set of high degrees and, bydefinition, D ( ≤ ∗ ) is the set of hyperimmune degrees. Similar ideas hold for the relations = ∗ and ∈ ∗ (as discussed in more detail below).My key observation is that this formalism has nothing to do with Turing computabilityper se. This motivates the following general definition.
Definition 1.2.1. A reduction concept is a triple ( X, ⊑ , x ) where X is a nonempty set, x ∈ X is some distinguished element and ⊑ is a partial pre-order on X . We also say thatthe pair ( ⊑ , x ) is a reduction concept on X . If ( X, ⊑ , x ) is a reduction concept, then for x, y ∈ X say that x is ⊑ - reducible to y if x ⊑ y and say that x is ⊑ - basic if it is ⊑ -reducibleto x .Let ( ⊑ , x ) be a reduction concept on X and R ⊆ X × X be a binary relation. Let ⊑ ↾ x = { y ∈ X | y ⊑ x } be the basic reals. Then define the bounding set for R as HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES B ⊑ ( R ) = { x ∈ X | ∃ y ⊑ x ∀ z ∈⊑ ↾ x [ z R y ] } and the non-dominating set for R as D ⊑ ( R ) = { x ∈ X | ∃ y ⊑ x ∀ z ∈⊑ ↾ x [ ¬ y R z ] } . Roughly, if we think of ⊑ is some sort of relative computability relation, then beingcomputable means computable from x and B ⊑ ( R ) is the set of elements of x ∈ X whichcompute an R -bound on the computable elements of X and D ⊑ ( R ) is the set of x ∈ X whichcompute an element which is not R -dominated by any computable element. If R is a relationgiving a notion of “small” and “big” sets as described above one can think of B ⊑ ( R ) as theset of elements computing a witness to the fact that the ⊑ -basic sets are small and D ⊑ ( R )as the set of elements computing a witness to the fact that the ⊑ -basic elements are not big. Example 1.2.2 ([12]) . Let x ∈ ω ω be some computable real, say the constant function at0. Then the pair ( ≤ T , x ) forms a reduction concept on the reals. The basic reals are thecomputable reals. For any binary relation R on the reals B ≤ T ( R ) is the set of Turing degreescomputing an element of X which R -bounds all the computable sets. Similarly D ≤ T ( R )is the set of Turing degrees computing an element of X which is not R -dominated by anycomputable set.The next example will be the central focus of the second half of this chapter. Example 1.2.3.
Let x ∈ ω ω be constructible. Then the pair ( ≤ L , x ) is a reductionconcept on ω ω where x ≤ L y if x ∈ L [ y ]. The basic reals are the constructible reals. Moregenerally, fix some inner model W ⊆ V and let ≤ W be constructibility relative to W . Thenif x ∈ ( ω ω ) W is any given real in W the pair ( ≤ W , x ) forms a reduction concept on Bairespace and the basic reals are those of W . Since this is the main case let me be explicit about HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES R be a relation on the reals of V . Theset B ≤ W ( R ) consists of all reals x in V such that in W [ x ] there is an R -bound on the realsof W . Similarly the set D ≤ W ( R ) consists of all reals x in V such that in W [ x ] there is a realwhich is not R -bounded by any real in W . For example, B ≤ W ( ≤ ∗ ) is the set of dominatingreals over W in V and D ≤ W ( ≤ ∗ ) is the set of unbounded reals over W in V .I will come back to this example in the next section. First, let me give some moreexamples of reduction concepts, though I will not treat them in detail in this thesis. Example 1.2.4.
Recall that for x, y ∈ P ( N ), the relation ≤ A is defined by x ≤ A y if andonly if x is definable in the standard model of arithmetic with an extra predicate for y . Thepair ( ≤ A , ∅ ) forms a reduction concept on P ( N ). In this case the basic reals are the setswhich are ∅ -definable in the standard model of arithmetic. More generally this could be donewith any model of PA . Example 1.2.5.
Recall that the relation of many-one polytime reduction, ≤ pm is defined by x ≤ pm y if and only if there is a function z which is computable in polynomial time such that n ∈ x if and only if z ( n ) ∈ y . The pair ( ≤ pM , ∅ ) is a reduction concept on P ( N ). Example 1.2.6.
Let κ > ω be an uncountable cardinal. Recently there has been much workin the descriptive set theory of “generalized” Baire and Cantor spaces, κ κ and 2 κ , includingvarious generalizations of cardinal characteristics of the continuum, see for instance [11].The same can be done in my framework for degrees of constructibility. For instance notionsof eventual domination, etc all make sense in the general context of κ κ and correspondingbounding and non-dominating sets can be constructed over the basic elements, ( κ κ ) L .The framework described above is flexible enough that ( X, ⊑ , x ) need not be some actualnotion of computability on the reals nor have an explicit relation to cardinal characteristicsof the continuum. For instance one might consider a class of models of a fixed theory HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES R onestudied, one would arrive at a diagram corresponding to when models with certain propertiesembed into one another. There are many possibilities, each giving a potentially interesting“Cicho´n diagram” of inclusions between the various bounding and non-dominating sets foran appropriate collection of relations. In future work I hope to explore all of these morefully.Presently however, let me restrict my attention to the types of cases described in the pre-ceding examples. Even in this general framework I can now prove a collection of implicationsgiving a version of the Cicho´n diagram. Theorem 1.2.7.
Let ( ⊑ , x ) be a reduction concept on ω ω extending ≤ T such that if x, y ⊑ z then x ◦ y ⊑ z . Interpreting arrows as inclusions, the implications in Figure 1.1 all hold. ∅ B ⊑ ( ∈ ∗ ) B ⊑ ( ≤ ∗ ) B ⊑ ( = ∗ ) D ⊑ ( = ∗ ) D ⊑ ( ≤ ∗ ) D ⊑ ( ∈ ∗ ) ω ω \ { x | x ⊑ x } Figure 1.1: A Cicho´n diagram for an arbitrary reduction concept on Baire space
Proof.
Note that slaloms can be computably coded by reals so, since the relation ⊑ extendsTuring computability the ∈ ∗ can be seen as a relation on the reals. I drop the ⊑ subscriptfor readability. Also, I’ll write “basic” for ⊑ -basic and if y ⊑ x then I’ll say that “ x builds y ”. The requirement that ⊑ be closed downwards under compositions will be used implicitlythroughout the argument where I will show that a function can build two other functionshence it can build their composition.Let’s begin with the easy cases, as shown in Figure 3. First I’ll show that B ( ∈ ∗ ) ⊆ B ( ≤ ∗ )or that every x building a slalom eventually capturing all the basic reals builds a real which HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES ∅ B ( ∈ ∗ ) B ( ≤ ∗ ) B ( = ∗ ) D ( = ∗ ) D ( ≤ ∗ ) D ( ∈ ∗ ) ω ω \ { x | x ⊑ x } Figure 1.2: The Easy Caseseventually dominates all basic reals. This is proved as follows. Suppose x ∈ B ( ∈ ∗ ) and let s ⊑ x be a slalom witnessing this. Then, define z ( n ) = max ( s ( n )) + 1. Notice that z ≤ T s so z ⊑ s and hence z ⊑ x . Moreover, since s eventually captures all basic reals, z musteventually dominate them all so x ∈ B ( ≤ ∗ ).Now let’s show that B ( ≤ ∗ ) ⊆ B ( = ∗ ) or that if x can build a real eventually dominatingall basic reals then it can build a real eventually different real from all basic reals. Noticehowever that once stated like this the proof simply the observation that if x eventuallydominates y and y + 1 (which is basic if y is since ≤ T is extended by ⊑ ), then, in particular x is eventually different from y .Next let’s show B ( ≤ ∗ ) ⊆ D ( ≤ ∗ ) or that if x builds a real which eventually dominates allbasic reals then it builds a real which is not dominated by any basic real. But now statedlike this it’s obvious.Next I show that D ( = ∗ ) ⊆ D ( ≤ ∗ ) or that if there is a real which is equal to everybasic real infinitely often then there is a real which is never dominated by any basic real.This is obvious though since if x were dominated by some basic real y then it could not beinfinitely-often-equal to the basic real y + 1.Now I show that D ( ≤ ∗ ) ⊆ D ( ∈ ∗ ) or that if x builds a real which is not dominated byany basic real then it builds a real that is never eventually captured by any basic slalom.Suppose x ∈ D ( ≤ ∗ ) and let y ⊑ x witness this. Then, if s is a basic slalom, let z be definedby z ( n ) is one plus the sum of the elements in s ( n ). Note that z ≤ T s so z is basic. Thus HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES n such that y ( n ) ≥ z ( n ) so y cannot be eventually captured by s .The last easy inclusion, that all the reals in every node are not themselves basic iscompletely straightforward. For instance, if x ∈ D ( ∈ ∗ ) is a real which is not eventuallycaptured by any basic slalom then of course x is not basic since if it were, the slalom n
7→ { x ( n ) } would be as well.Now I move on to the more difficult inclusions, starting with B ( ∈ ∗ ) ⊆ D ( = ∗ ). Substan-tively this states that if a real x builds a slalom eventually capturing all basic functionsthen x also builds a real which is infinitely-often-equal to all basic functions. In fact I willshow a more general claim that implies this. The following lemma and proof is essentially areinterpretation of Theorem 1.5 from [5]. Lemma 1.2.8.
For any real x the following are equivalent.1. There is a real y ⊑ x such that for all basic z ∈ ω ω , there exist infinitely many n ∈ ω such that y ( n ) = z ( n )
2. There is a basic z ∈ ω ω and a z -slalom s ⊑ x such that for all basic y ∈ ω ω there areinfinitely many n ∈ ω such that y ( n ) ∈ s ( n ) .Moreover, given an infinitely-often-equal real as in , one can build from it a z -slalom as in and given a z -slalom s as in one can build an infinitely-often-equal real as in . Thus, x ∈ D ( = ∗ ) if and only if there is a basic z ∈ ω ω and a z -slalom which captures each of thebasic reals infinitely often. Before proving Lemma 1.2.8, notice that it implies the inclusion B ( ∈ ∗ ) ⊆ D ( = ∗ ) sinceany slalom which captures every basic real cofinitely often must in particular capture eachbasic real infinitely often so if x ∈ B ( ∈ ∗ ) builds such a slalom, by the lemma x must be ableto build an infinitely-often-equal real as well. HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES Proof of Lemma 1.2.8.
The forward direction is obvious: suppose that y is an infinitely-often-equal real. Then clearly the 1-slalom s : ω → [ ω ] such that s ( n ) = { y ( n ) } is ≤ T -computable from y and hence ⊑ -reducible to y , thus giving the desired z -slalom.For the backward direction fix a basic real z such that there exists a z -slalom as in thestatement of 2. I need to find a real y which is infinitely often equal to every basic real. Ina basic fashion, fix a family of finite, nonempty, pairwise disjoint subsets of ω enumerated { J n,k | n < ω & k ≤ z ( n ) } which collectively cover ω . Since z is assumed to be basicthere is no problem building such a partition, for example one could use singletons. Label J n = S k ≤ z ( n ) J n,k . Then for each basic v ∈ ω ω let v ′ : ω → ω <ω be the function defined by v ′ ( n ) = f ↾ J n . More generally let J = { v : ω → ω <ω | dom( v ( n )) = J n } . Notice thatthe basic elements of J are exactly { v ′ | v ∈ ω ω & v ⊑ } since from any v ′ we can build v and vice versa (by the the fact that the J n ’s are basic). But now since the v ′ ’s are basicand each one codes a real one can by applying 2 plus some simple coding to find a z -slalom, s : ω → ( ω <ω ) <ω such that for every n ∈ ω | s ( n ) | ≤ z ( n ) and s ( n ) is a set of finite partialfunctions from J n to ω and for every basic v ′ ∈ J there are infinitely many n ∈ ω such that v ′ ( n ) ∈ s ( n ).Let me denote s ( n ) = { w n , ..., w nz ( n ) } . Now set y n = S k ≤ z ( n ) w nk ↾ J n,k and let y = S n<ω y n .Notice that this gives an element of ω ω since the J n,k ’s were disjoint and collectively covered ω . I claim that y is as needed. Clearly y is reducible to the J n,k ’s, which are basic, and the w nk ’s, which are reducible to s so y is reducible to s . It remains to see that it is an infinitely-often-equal real. To see this, let v ∈ ω ω be basic and fix some n such that v ′ ( n ) ∈ s ( n )(recall that there are infinitely many such n ). Notice that since v ′ ( n ) ∈ s ( n ) there must besome k ≤ z ( n ) such that v ↾ J m = w nk . Now let x n ∈ J n,k (recall that this set is assumed tobe non-empty). We have that v ( x n ) = w nk ( x n ) = g ( x n ). But there are infinitely many such n and hence infinitely many such x n so this completes the proof. HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES
18A similar proof produces the last inclusion, B ( = ∗ ) ⊆ D ( ∈ ∗ ). In words this inclusion statesthat any real which can build a real which is eventually different from all basic reals canbuild a real which is not eventually captured by any given slalom. I will prove the followingmore general lemma, whose statement and proof is inspired by [5], Theorem 2.2. Given a z -slalom s and a function x let me say that x is eventually never captured by s if there issome k such that for all l > k we have x ( l ) / ∈ s ( l ). Lemma 1.2.9.
For any real x , the following are equivalent.1. The real x is eventually different from all basic reals.2. The real x is such that for all basic reals z and all basic z -slaloms s for all but finitelymany n ∈ ω x ( n ) / ∈ s ( n ) .Therefore x ∈ B ( = ∗ ) if and only if x builds a real which is eventually never captured by anybasic z -slalom for any basic z . Let me note before I prove Lemma 1.2.9 that it proves the inclusion B ( = ∗ ) ⊆ D ( ∈ ∗ ) andhence Theorem 1.2.7. To see why, suppose that x ∈ B ( = ∗ ) and, without loss of generalitysuppose that x itself is a real which is eventually different from all basic reals. Then bythe lemma x is eventually never captured by any basic slalom so, in particular for infinitelymany n x ( n ) / ∈ s ( n ) for all basic s , which means x ∈ D ( ∈ ∗ ). Proof of Lemma 1.2.9.
Fix some x ∈ ω ω . The backward direction of this lemma is easy: if x is eventually never captured by any basic z -slalom for any basic z then in particular it iseventually never captured by the slalom sending n
7→ { y ( n ) } for each basic y and hence itis eventually different from each basic y .For the forward direction, assume x is eventually different from all basic functions. Fixa basic z and, like in the proof of Lemma 1.2.8, in a basic fashion partition ω into finite,disjoint, non-empty sets { J n,k | k ≤ z ( n ) } . Let J n = S k ≤ z ( n ) J n,k . Let x ′ : ω → ω <ω be the HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES x ′ ( n ) = x ↾ J n . Then if s is any basic z -slalom, let s ′ be such that oninput n gives z ( n ) many finite partial functions w n , ..., w nz ( n ) with domain J n where for all k ≤ z ( n ) and l ∈ J n w nk ( l ) is the k th greatest number in the set s ( l ) if such exists and 0(say) otherwise. Suppose now towards a contradiction that there is a basic z -slalom s suchthat x ( n ) ∈ s ( n ) for infinitely many n . For each n let s ′ ( n ) = { w n , ..., w nh ( n ) } . Then define y n = S k ≤ z ( n ) w nk ↾ J n,k and let y = S n<ω y n . Clearly y can built using s , the function z andthe J n,k ’s each of which is basic so y is basic. Thus there is a k such that for all n > k wehave that x ( n ) = y ( n ). But, since there are infinitely many n such that x ( n ) ∈ s ( n ), thereare infinitely many n > k such that x ( n ) ∈ s ( n ) and therefore it follows that similarly wemust have that there are infinitely many n > k such that x ′ ( n ) agrees with some w nj on someelement of their shared domain for some j ≤ z ( n ). But this means x ( k ) = y ( k ) for some k ∈ J n,j for infinitely many n ’s and j ’s which is a contradiction.Since this was the final inclusion to prove, Theorem 1.2.7 is now proved as well.Thus, even in this broad context one can construct diagrams for a wide variety of reduc-tion concepts and a correspondence starts to form with the Cicho´n diagram. This extendsthe proof given in the case of Turing degrees in [12] and gives a good framework for inves-tigations into various computability reduction concepts. What it does not show, however,is that any of these nodes are non-empty or that the inclusions are strict. Indeed this isnot necessarily the case. For instance B ≤ T ( ∈ ∗ ) = B ≤ T ( ≤ ∗ ) (see [12]). This is because, by atheorem of Rupprecht, the set B ≤ T ( ∈ ∗ ) is simply the high reals, which as I mentioned aboveis also B ( ≤ ∗ ). The analogue of this fact in the case of the classical Cicho´n diagram is falsesince add( N ), the analogue of B ≤ T ( ∈ ∗ ), can consistently be less than b , the analogue of B ( ≤ ∗ ). The authors of [12] take this as evidence that the ≤ T -Cicho´n diagram provides “onlyan analogy, not a full duality” [12, p. 3] with the classical Cicho´n diagram. Theorem 1.2.7proves the existence of a wide variety of such diagrams, therefore raising the question in each HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES ≤ T diagram gives only an analogy, I will show in the next section that in the ≤ W diagram the inclusions proved in Theorem 1.2.7 constitute the only ones true in every modelof ZFC , thereby suggesting something closer to a true duality. ≤ W From now on fix an inner model W | = ZFC . I work in the language of set theory with an extrapredicate for W and the theory ZFC( W ), that is ZFC with replacement and comprehensionholding for formulas containing W . I view W = L as a central case but it turns out that theanalysis works out the same for arbitrary W .Before presenting the full ≤ W -Cicho´n diagram, let me state clearly what the boundingand dominating sets are for the combinatorial relations defined in the last section for ≤ W .1. B ( ∈ ∗ ) is the set of reals x such that there is a slalom s ∈ W [ x ] that eventually capturesall reals in W .2. B ( ≤ ∗ ) is the set of reals x such that there is a real y ∈ W [ x ] that eventually dominatesall reals in W . These are sometimes called dominating reals (for W ).3. B ( = ∗ ) is the set of reals x such that there is a real y ∈ W [ x ] that is eventually differentfrom all reals in W . These are sometimes called eventually different reals (for W ).4. D ( ∈ ∗ ) is the set of reals x such that there is a real y ∈ W [ x ] that is not eventuallycaptured by any slalom in W .5. D ( ≤ ∗ ) is the set of reals x such that there is a real y ∈ W [ x ] that is not eventuallydominated by any real in W . These are sometimes called unbounded reals (for W ). HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES D ( = ∗ ) is the set of reals x such that there is a real y ∈ W [ x ] that is equal infinitelyoften to every real in W . These are sometimes called infinitely-often-equal reals (for W ).In this section I will study how a variety of known forcing notions over W can createseparations in the ≤ W -Cicho´n diagram as described in the previous section. Of courseZFC( W ) cannot prove any separations since if V = W or, more generally V and W havethe same reals, every node in the ≤ W -diagram will be empty. However, using simple forcingnotions I will show that one can produce a wide variety of possible constellations for the ≤ W -diagram. The main theorem of this section is the following. Theorem 1.3.1.
The Cicho´n diagram for ≤ W as described in the previous section is completefor ZFC( W )-provable implications. In other words if A and B are two nodes in the diagramand A ⊆ B does not follow from the transitive closure of the arrows in the ≤ W -diagram thenthere is a forcing extension of W where A * B . That these implications all hold follows from the main theorem of the previous sectionsince ≤ W extends ≤ T . ∅ B ≤ W ( ∈ ∗ ) B ≤ W ( ≤ ∗ ) B ≤ W ( = ∗ ) D ≤ W ( = ∗ ) D ≤ W ( ≤ ∗ ) D ≤ W ( ∈ ∗ ) ω ω \ ( ω ω ) W Let me note one word on the relation between my diagram and the standard Cicho´ndiagram as commonly studied, for example in [7]. Here I have focused on the so-calledcombinatorial nodes as discussed by [12]. As noted in the previous section, I view mydiagram in correspondence with the classical one via the mapping sending unbounded or
HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES x such that in W [ x ] the reals of W are not unbounded or dominating. I have included this fragment of theCicho´n diagram to make this analogy clear visually. ℵ add( N ) b non( M ) cov( M ) / d ( = ∗ ) d cof( N ) 2 ℵ Figure 1.3: The Combinatorial Nodes of the Standard Cicho´n DiagramThe details of these correspondences for ≤ T can be found in [12] and similar ideas hold inthe present case, with one exception: cov( M ) / d ( = ∗ ). As noted in Fact 1.1.3 these cardinalsare the same, however Zapletal has shown in [51] that their degree theoretic analogues arein fact different, thus solving a well known problem of Fremlin. I will mention Zapletal’stheorem again at the end of this chapter in connection with extensions of the current work. The first forcing I will look at is Sacks forcing, S . Recall that conditions in S are perfect trees T ⊆ <ω ordered by inclusion. If G is S -generic then the unique branch in the intersectionof all members of G is called a Sacks real . I denote such a real s . Theorem 1.3.2.
In the Sacks extension all nodes of ≤ W -Cicho´n diagram other than ω ω \ ( ω ω ) W are empty.Proof. Recall that Sacks forcing has the Sacks property, Definition 1.1.5, see Lemma 7.3.2of [7]. As a result, all reals added by S and hence all reals in V that are not in W can becaptured by a slalom from the ground model i.e. W . Thus, W [ s ] thinks that D ( ∈ ∗ ) is empty HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES ∅ B ( ∈ ∗ ) B ( ≤ ∗ ) B ( = ∗ ) D ( = ∗ ) D ( ≤ ∗ ) D ( ∈ ∗ ) ω ω \ ( ω ω ) W Figure 1.4: After Sacks forcingbut s ∈ ω ω \ ( ω ω ) W and hence the only non-empty set in the ≤ W -Cicho´n diagram is thelatter. Let C = Add ( ω,
1) be the forcing to add one Cohen real. The main theorem of this sectionis:
Theorem 1.3.3.
Let c be a Cohen real generic over W . Then in W [ c ] the following hold:1. ∅ = B ( ∈ ∗ ) = B ( ≤ ∗ ) = B ( = ∗ ) D ( = ∗ ) = D ( ≤ ∗ ) = D ( ∈ ∗ ) = { x | ∃ c ∈ W [ x ] Cohen over W } = ω ω \ ( ω ω ) W ∅ B ( ∈ ∗ ) B ( ≤ ∗ ) B ( = ∗ ) D ( = ∗ ) D ( ≤ ∗ ) D ( ∈ ∗ ) ω ω \ ( ω ω ) W Figure 1.5: After Cohen forcing
HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES Proof.
There are two parts to this proof. First I need to show that all of the elements onthe left are empty. Since B ( ∈ ∗ ) ⊆ B ( ≤ ∗ ) ⊆ B ( = ∗ ) it suffices to show that Cohen forcingadds no reals which are eventually different from all ground model reals. This is a standardargument but I repeat it here for completeness, see also [10, p. 83]. Let { p i } i ∈ ω enumeratethe conditions of C and suppose that (cid:13) C ˙ x : ω → ω . Then, for each i , pick a q i ≤ p i whichdecides the value of ˙ x ( i ) in other words let q i (cid:13) ˙ x (ˇ i ) = ˇ j i for some j i . Now, in the groundmodel, set y ( i ) = j i . Finally, suppose for contradiction that there was a k ∈ ω and a p ∈ C such that p (cid:13) ∀ l > ˇ k ˇ y ( l ) = ˙ x ( l ). But then one can find an i > k and a q i ≤ p such that q i (cid:13) ˇ y ( i ) = ˙ x ( i ), which is a contradiction.So Cohen forcing leaves the left side of the diagram trivialized. The right side howeverchanges since it’s dense for c to equal every real in W infinitely often so c ∈ D ( = ∗ ). Thesecond part of the proof is to show that every real added by Cohen forcing adds an element to D ( = ∗ ). Since D ( = ∗ ) ⊆ D ( ≤ ∗ ) ⊆ D ( ∈ ∗ ) ⊆ ω ω \ ( ω ω ) W it suffices to show that, ω ω \ ( ω ω ) W ⊆D ( = ∗ ). Let x ∈ W [ c ] \ W be a new real and consider now the model W [ x ]. By theintermediate model theorem it must be the case that W [ x ] is a generic extension of W andthat W [ c ] is a generic extension of W [ x ] so the forcing to add x is a non trivial factor ofCohen forcing so it must in fact be isomorphic to it by Theorem 3.3.1 of [7]. Thus in W [ x ]there is a real d which is Cohen generic over W , and d is infinitely often equal to every realin W so x ∈ D ( = ∗ ). I denote random real forcing by B . The diagram for random real forcing is as described inthe theorem below and can be proved in a very similar way to that of Cohen forcing usingthe standard facts found in [7, Chapter 3]. Theorem 1.3.4.
Let r be a random real over W . Then in W [ r ] the ≤ W -Cicho´n diagram is HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES determined by the separations B ( ∈ ∗ ) = B ( ≤ ∗ ) = D ( = ∗ ) = D ( ≤ ∗ ) = ∅ and B ( = ∗ ) = D ( ∈ ∗ ) = ω ω \ ( ω ω ) W . ∅ B ( ∈ ∗ ) B ( ≤ ∗ ) B ( = ∗ ) D ( = ∗ ) D ( ≤ ∗ ) D ( ∈ ∗ ) ω ω \ ( ω ω ) W Figure 1.6: After Random Real forcingThe proof of this theorem follows from the following list of facts that are well known andcan be found in [7], Chapter 3.
Fact 1.3.5.
The random real forcing B
1. Adds no unbounded reals,2. Adds an eventually different real and3. If x ∈ W [ r ] ∩ ω ω \ W ∩ ω ω then there is a real which is random over W in W [ x ] .Proof of Theorem 1.3.4. Since by 1 of Fact 1.3.5, B adds no unbounded reals D ( ≤ ∗ ) is empty.Now, suppose x ∈ W [ r ] \ W , then there is a y ≤ W x which is also random over W by 3 ofFact 1.3.5. Thus by 2 of Fact 1.3.5 we get that x ∈ B ( = ∗ ). Therefore ω ω \ ( ω ω ) W ⊆ B ( = ∗ )and the result follows. Let me now turn to Laver forcing, L . Recall that conditions in Laver forcing are trees T ⊆ ω <ω with a distinguished stem , that is, a linearly ordered initial segment, after which HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES L form a real, called a Laver real. Let l denote such a real over W .Recall that l is dominating. The main theorem of this section is Theorem 1.3.6.
Let l be a Laver real over W . Then in W [ l ] we have that ∅ = B ( ∈ ∗ ) = D ( = ∗ ) and all other nodes are equal to the set of all new reals. ∅ B ( ∈ ∗ ) B ( ≤ ∗ ) B ( = ∗ ) D ( = ∗ ) D ( ≤ ∗ ) D ( ∈ ∗ ) ω ω \ ( ω ω ) W Figure 1.7: After Laver forcingAs before this theorem follows from well known facts about L . In particular the Laverproperty, Definition 1.1.7, which holds of L [7, Theorem 7.3.29], implies that there are noinfinitely often equal reals in W [ l ] . Thus it suffices to note that l is dominating and, by [28,Theorem 7], that Laver reals satisfy the following minimality property: if x is a real suchthat x ∈ W [ l ] \ W then l ∈ W [ x ]. Therefore every new real constructs a dominating real,hence the equality between B ( ≤ ∗ ) and ω ω \ ( ω ω ) W . Next I look at is Miller’s rational perfect tree forcing, PT . Recall that PT is the set of perfecttrees T ⊆ ω <ω so that for all s ∈ T there is a t ⊇ s with ω -many immediate successors. Theorder is inclusion and the unique branch through the trees in the generic is called a Millerreal . Let us denote such a real by m . I would like to thank Professor Martin Goldstern who explained this fact to me on Mathoverflow,https://mathoverflow.net/questions/287977/does-laver-forcing-add-an-infinitely-often-equal-real .
HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES Theorem 1.3.7.
Let m be a Miller real over W . Then the ≤ W diagram in W [ m ] is de-termined by ∅ = B ( = ∗ ) = D ( = ∗ ) and all other nodes are equal to the set of all new reals. ∅ B ( ∈ ∗ ) B ( ≤ ∗ ) B ( = ∗ ) D ( = ∗ ) D ( ≤ ∗ ) D ( ∈ ∗ ) ω ω \ ( ω ω ) W Figure 1.8: After rational perfect tree forcingThis is proved in the same way as for Laver forcing. It suffices to note that PT adds noeventually different real, see [7, Theorem 7.3.46, Part 1], PT adds no infinitely often equalreal as it enjoys the Laver property ([7, Theorem 7.3.45]) and m is of minimal degree, see[28, Theorem 3]. Let D be Hechler forcing and let d be the associated dominating real. Recall that conditionsof D are pairs ( p, F ) where p is a finite partial function from ω to ω and F is a finite familyof elements of ω ω . The order is given by ( q, G ) ≤ D ( p, F ) if and only if q ⊇ p , G ⊇ F andfor all n ∈ dom( q ) \ dom( p ) and all x ∈ F , q ( n ) > x ( n ). Note that since d is dominating, d ∈ B ( ≤ ∗ ). Theorem 1.3.8.
After Hechler forcing over W the ≤ W -diagram has1. ∅ = B ( ∈ ∗ ) ,2. B ( ≤ ∗ ) = B ( = ∗ ) and HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES D ( = ∗ ) = D ( ≤ ∗ ) = D ( ∈ ∗ ) = ω ω \ ( ω ω ) W . ∅ B ( ∈ ∗ ) B ( ≤ ∗ ) B ( = ∗ ) D ( = ∗ ) D ( ≤ ∗ ) D ( ∈ ∗ ) ω ω \ ( ω ω ) W Figure 1.9: After Hechler forcingThe proof of this theorem is broken up into several lemmas. First I show that D addsno slaloms eventually capturing all the ground model reals. This is well known but includedhere for completeness. Lemma 1.3.9.
After Hechler forcing over W the set B ( ∈ ∗ ) is empty.Proof. Let me begin with a simple observation about Hechler forcing: if s is a sentence in theforcing language and p is the stem of a condition (the first coordinate) then it cannot be thatthere are there are finite families of functions F and G such that ( p, F ) (cid:13) s and ( p, G ) (cid:13) ¬ s .To see why, simply notice that ( p, F ∪ G ) is a condition extending them both. Now, usingthe weak homogeneity of Hechler forcing, suppose that (cid:13) D “ ˙ s is a slalom eventually capturingall elements of ( ω ω ) W ”. Now fix an enumeration of ω <ω = { p , p , p , ... } and consider thefollowing function x : ω → ω such that x ( n ) = sup { k | ∃ i < n ∃F ( p i , F ) (cid:13) ˇ k ∈ ˙ s ( n ) } + 1.Note that x is definable in W . Claim 1.3.10.
The function x is total and well defined.Proof. To see this, notice that since the maximal condition forces that ˙ s names a slalom, allconditions force that for all n , ˙ s ( n ) has size at most n . In particular, no condition can force HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES n check names to be in s ( n ). Moreover, by the simple observation I began with,there cannot be more than n check names forced to be in ˙ s by any set of conditions sharingthe same stem. Thus, since there are only finitely many stems being considered, each ofwhich can only be paired to force at most n check names, there are at most n numbers inthe set { k | ∃ i < n ∃F ( p i , F ) (cid:13) ˇ k ∈ ˙ s ( n ) } so x is well defined and always finite.Now work in W [ d ]. It remains to show that x is not eventually captured by the slalom s = ˙ s d . Suppose not and let k, j ∈ ω such that ( p j , F ) (cid:13) ∀ l > ˇ k ˇ x ( k ) ∈ ˙ s (ˇ k ). Let nowlet l > k, k be such that ( p l , G ) ≤ ( p j , F ). Then, ( q, G ) (cid:13) ˇ x ( l ) ∈ ˙ s ( l ) but this implies x ( l ) ≥ x ( l ) + 1, which is a contradiction.Continuing, recall the following theorem of Brendle and L¨owe. I have adapted it to ourspecific situation and terminology: Theorem 1.3.11. ([13, Corollary 13]) If d is Hechler generic over W and x ∈ W [ d ] ∩ ω ω iseventually different from every y ∈ W ∩ ω ω , then x eventually dominates every y ∈ W ∩ ω ω . Therefore all the reals in W [ d ] in B ( = ∗ ) are automatically in B ( ≤ ∗ ). As an immediatecorollary the following is true. Corollary 1.3.12.
In the extension of W by a Hechler real, B ( = ∗ ) = B ( ≤ ∗ ) . Thus, we know what happens on the left side of the diagram. For the right side of thediagram, the following fact is well known and easily verified:
Fact 1.3.13.
Let d be D -generic over W . Then d mod 2 i.e. the parity of d is a Cohengeneric over W . Therefore Hechler forcing adds Cohen reals. Indeed, since C ∗ ˙ C is forcing equivalent to C , by the intermediate model theorem D can be decomposed in to C ∗ Q where Q is somequotient forcing. But then D ∼ = C ∗ ˙ Q ∼ = C ∗ ˙ C ∗ ˙ Q ∼ = C ∗ ˙ D . So Hechler forcing is the same HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES ∅ for B ( ∈ ∗ ), the dominating reals, for B ( ≤ ∗ ) = B ( = ∗ ) andthe reals that add Cohen real, which by the previous section, are all included in D ( = ∗ ) andthus the entire right column. To finish the analysis, I use the following fact, due to Palumbo. Fact 1.3.14. ([42, Theorem 8.1]) Let d be D -generic over W and let M be an intermediatemodel i.e. W ⊆ M ⊆ W [ d ] . Then if M = W , there is a real x ∈ M which is Cohen-genericover W . Using Fact 1.3.14 I can now show the following:
Corollary 1.3.15.
In the extension of W by a Hechler real, all the new reals construct areal which is equal to the reals in W infinitely often i.e. ω ω \ ( ω ω ) W = D ( = ∗ ) .Proof. Let x ∈ W [ d ] \ W be a real. Then by Fact 1.3.14 there is C -generic real over W in W [ x ] so x ∈ D ( = ∗ ).Notice that this completely determines the diagram for a Hechler real. Since D ( = ∗ ) isall new reals, every node in the diagram is a subset of it. Thus all nodes on the right sideare equal, B ( ∈ ∗ ) is empty and B ( ≤ ∗ ) = B ( = ∗ ) form a proper subset of the D ’s. This finishesthe proof of Theorem 1.3.8 Let E be eventually different forcing , which is defined like D except that stems of extensionsneed simply be eventually different from the reals in the second component, not dominating. HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES
31I will show that:
Theorem 1.3.16.
Assume that every set of reals in L ( R ) has the Baire property (this isimplied by sufficiently large cardinals). Let e be an E -generic real over W . Then in W [ e ] thefollowing hold:1. B ( ∈ ∗ ) = B ( ≤ ∗ ) = ∅ ,2. B ( = ∗ ) ( D ( = ∗ ) = D ( ≤ ∗ ) = D ( ∈ ∗ ) = ω ω \ ( ω ω ) W .Thus in particular the full diagram for eventually different forcing is as shown in Figure1.3.16. ∅ B ( ∈ ∗ ) B ( ≤ ∗ ) B ( = ∗ ) D ( = ∗ ) D ( ≤ ∗ ) D ( ∈ ∗ ) ω ω \ ( ω ω ) W Figure 1.10: After Eventually Different forcingTo prove this I will use a series of lemmas similar to those used in the case of Hech-ler forcing. First, a straightforward modification of Lemma 1.3.9 shows that there are nodominating reals in W [ e ]: Lemma 1.3.17. W [ e ] | = B ( ∈ ∗ ) = B ( ≤ ∗ ) = ∅ To complete the analysis of the ≤ W -diagram after forcing with E , I need the analogy ofPalumbo’s Fact 1.3.14 for E . Unfortunately, his argument uses a tree version of D that, asfar as I can tell, is not available for E . As such, I only know how to prove Palumbo’s resultfor E assuming sufficient large cardinals. I conjecture that it should hold in ZFC . HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES Lemma 1.3.18.
Assume that every set of reals in L ( R ) has the property of Baire. Then inevery nontrivial intermediate model between W and W [ e ] there is a real c which is C -genericover W . A proof of this is sketched in [42, pg 38] for D but the reader will notice that it goesthrough equally well for E . Indeed the centerpiece of the argument involves a fact, due toShelah and Gitik [26, Proposition 4.3] that given any sufficiently well-defined σ -centeredforcing P , if certain filters of P in L ( R ) have the property of Baire, then P will add a Cohenreal. It is not hard to see from the combination of the Gitik-Shelah and the Palumboarguments that “sufficiently well defined” includes all subforcings of E . Thus, assuming allsets of reals have the property of Baire the result goes through.Using this lemma, by the same argument given for D , we have the proof of Theorem1.3.16.The use of large cardinals here is unfortunate and I hope it can be improved on. Let menote however that even without large cardinals I have shown that there is a model realizingthe cut determined by B ( ∈ ∗ ) = B ( = ∗ ) = ∅ . In this section I study
Localization forcing , the forcing to add a generic slalom capturing allground model reals.
Definition 1.3.19 (Localization Forcing (cf [13])) . The localization forcing
LOC is definedas the set of pairs ( s, F ) such that s ∈ ([ ω ] <ω ) <ω is a finite sequence with | s ( n ) | ≤ n forall n < | s | and F is a a finite family of functions in Baire space with |F | ≤ | s | . The orderis ( t, G ) ≤ LOC ( s, F ) if and only if t ⊇ s , G ⊇ F and x ( n ) ∈ t ( n ) for all x ∈ F and all n ∈ | t | \ | s | .Intuitively we think of the first component as a finite approximation to a slalom we are HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES s ). The second component is the set of functions we are promising to capturefrom that stage onwards.Unfortunately I do not have a full characterization of the diagram in the case of LOC .The following theorem summarizes the state of knowledge.
Theorem 1.3.20.
Let s be a slalom which is LOC -generic over W . Then in W [ s ] all thenodes in the diagram are nonempty (with the exception of ∅ ) and we have that B ( ∈ ∗ ) is aproper subset of B ( ≤ ∗ ) and D ( = ∗ ) . Also B ( ≤ ∗ ) ( B ( = ∗ ) and D ( ≤ ∗ ) ( D ( ∈ ∗ ) . In particular,Figure 1.11 is a partial diagram for LOC . ∅ B ( ∈ ∗ ) B ( ≤ ∗ ) B ( = ∗ ) D ( = ∗ ) D ( ≤ ∗ ) D ( ∈ ∗ ) ω ω \ ( ω ω ) W ??Figure 1.11: Partial diagram after Localization forcingProving this theorem amounts to showing that LOC adds B , D and E generics. I startwith D . Notice first that LOC adds a dominating real. Indeed if s is a generic slalom in W LOC then d ( n ) := max s ( n ) has this property. This is actually a Hechler real: Lemma 1.3.21.
Let s ∈ W LOC be a generic slalom eventually capturing all ground modelreals. Then, d ( n ) := max s ( n ) is D -generic over W . To prove this I will need a simplified version of D : in the first component of a condition Iwill assume that the domain is a finite initial segment of ω and instead of having the secondcomponent of a condition of D be a finite family of functions, it will be a single function. HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES q, y ) ≤ D ( p, x ) if and only if q extends p , for all n ∈ dom( q ) \ dom( p ), q ( n ) ≥ x ( n )and for all n ∈ ω , and y ( n ) ≥ x ( n ). It’s not hard to see that this version of D is forcingequivalent to the original one I defined. Proof.
Recall that a projection π : P → Q between two posets is an order preserving mapwhich sends the maximal element of P to the maximal element of Q and for all p ∈ P and all q ≤ π ( p ) there is some p ≤ p such that π ( p ) ≤ q . If a projection exists between P and Q thenthe image π ′′ G of a P -generic filter generates a Q -generic filter. Therefore to prove the lemmait suffices to show that the map π : LOC → D such that π ( s, F ) = ( n max s ( n ) , Σ F )where Σ F is the pointwise sum, is a projection. To see why, note that if ( s, F ) ∈ LOC and letting, for all n ∈ dom( s ), p ( n ) = max s ( n ) and x = Σ F , then the pair ( p, x ) is a D condition and the union of all conditions such defined from elements of the LOC genericdefining s is the d from the statement of the lemma.It is routine to check that π (1 LOC ) = 1 D and that the map π is order preserving. Thedifficulty is in verifying the third condition of projections. To this end, let ( s, F ) ∈ LOC and let ( p, x ) = π ( s, F ). Let ( p ′ , x ′ ) ≤ ( p, x ) and let D ⊆ D be a set of conditions whichis dense below ( p ′ , x ′ ). It suffices to find a strengthening ( t, G ) of ( s, F ), such that ( n max t ( n ) , Σ G ) ∈ D . To do this, let ( q, z ) ∈ D strengthen ( p ′ , x ′ ) so that | dom( q ) | > | dom( s ) | + 2.Now, we can build our new LOC condition. Define H : ω → ω by H ( n ) = z ( n ) − x ( n ).Notice that since x ′ ( n ) was assumed to be bigger than x ( n ) for all n and z ( n ) ≥ x ′ ( n ) sinceit is a strengthening it follows that H is in fact always nonnegative. Moreover, x + H =Σ F + H = z . It remains to show that there is a t ⊇ s such that dom( t ) = dom( q ),for all n ∈ dom( t ), max t ( n ) = q ( n ) and for all n ∈ dom( t ) \ dom( s ) and all v ∈ F , v ( n ) ∈ t ( n ). Once this has been done ( t, F ∪ { H } ) will be the desired condition. I claimthat this is all possible. I will describe a t extending s be defined on the domain of q (by HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES q contains that of s ). Since | dom( q ) | > | dom( s ) | + 2, the domainof t will be large enough to accommodate the side condition F ∪ { H } . Let |F | = k andenumerate F = { v , ..., v k − } . Note that k < n for all n ∈ dom( q ) \ dom( s ). Now, for each n ∈ dom( q ) \ dom( s ), let me define t ( n ). Notice first that one must put in all k numbers { v ( n ) , ..., v k − ( n ) } and we also want max t ( n ) = q ( n ) so add this in too. Since n > k ,one may add up to n − k − { j , ..., j n − k − } such that each one is lessthan q ( n ) and different from all numbers in the set { v ( n ) , ..., v k − ( n ) , q ( n ) } . Let t ( n ) bethis set plus any of the additional numbers that fit. Note that the definition of LOC allows | t ( n ) | ≤ n so we do not need to meet this bound everywhere. What matters is that, sinceby construction q ( n ) ≥ x ′ ( n ) for all n / ∈ dom( p ) and x ′ ( n ) ≥ Σ i The forcing LOC adds an E -generic real.Proof. Given a condition ( s, F ) ∈ LOC define a stem for an E -condition as p s : dom( s ) → ω by letting for all n ∈ dom( s ) p s ( n ) be equal to the k th natural number m not in the set s ( n )where the pointwise sum Σ s ( n ) ≡ k mod n . I claim that the map π : LOC → E definedby π ( s, F ) = ( p s , F ) is a projection. Clearly the maximal condition is sent to the maximalcondition and this map is order preserving. Let ( s, F ) ∈ LOC , and let ( q, G ) ≤ E ( p s , F ). Weneed to show that there is a strengthening of ( q, G ) in the image of π . To this end, note thatwe can assume with out loss that |G| < dom( q ) since otherwise we can strengthen to makethis true. Now, define a partial slalom as follows: s q : dom( q ) → [ ω ] <ω . For n ∈ dom( p )let s q ( n ) = s ( n ). For n / ∈ dom( p ) let q ( n ) = m and suppose that m is the k th not in { x ( n ) | x ∈ F } and suppose that this set has size l < n (the < follows from the fact that( p, F ) is in the image of π ). Then, pick n − l numbers m l , m l +1 , ..., m n − all greater than HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES x ( n ) for x ∈ F and not equal to m so that Σ x ∈F x ( n ) + Σ n − i = l m i ≡ k mod n . This canbe accomplished, for instance, as follows: if Σ x ∈F x ( n ) ≡ j mod n then let m l ≡ k − j mod n greater than all the f ( n )’s and let all other m i ’s be multiples of n . Finally let s q ( n ) = { x ( n ) | x ∈ F } ∪ { m l , ..., m n − } . Then ( s q , G ) ≤ ( s, F ) and π ( s q , G ) = ( q, G ) as needed.Finally, Lemma 1.3.23. Any forcing adding a slalom eventually capturing all ground model realsadds a random real. In particular LOC adds a random real.Proof. By Corollary 3.2 of [35] adding a slalom eventually capturing all ground model realsis equivalent to adding a Borel null set which covers all Borel null sets coded in the groundmodel. Let N ⊆ ω ω be such a null set and let y / ∈ N . Then y is not in any ground modelnull set so y is a random real.Combining all of these results then proves Theorem 1.3.20 since both D and E add Cohenreals realizing the split down the middle in Figure 1.11 and B adds a bounded real not caughtin any old slalom so D ( ≤ ∗ ) is strictly contained in D ( ∈ ∗ ).As an aside notice that there seem to be other eventually different reals added by LOC : Observation 1.3.24. Let s ∈ W LOC be a generic slalom eventually capturing all groundmodel reals. Let a ( n ) be defined as the least k / ∈ s ( n ) . Then a is a real which is eventuallydifferent from all ground model reals but is not an E -generic real.Proof. First notice that the a described in the theorem is in fact eventually different fromall ground model reals since every real eventually is captured by s and after that point a isdifferent from it. Moreover, notice that a is not only not dominating over the ground modelreals but actually not even unbounded since, given any real f ∈ W growing faster than theidentity ( n n + 2 even), the least k not in s ( n ) must be less than f ( n ) since | s ( n ) | = n .From this it follows that a is not an E -generic real since it is not unbounded. HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES a is or if it is a previously studied notion. In particular, I don’t know if thisreal is random over W , though I conjecture that it is. Let me finish this section by noting that it follows from what I have shown that the ZFC( W )-provable subset implications implied by Theorem 1.2.7 are the only ones. In other words,Theorem 1.3.1 is proved. Indeed a simple inspection of the diagrams above show that everyimplication shown in Figure 1 is consistently strict and no other implications are true inevery V extending W . This shows also that the analogue discussed in the previous sectionholds in a robust way with the traditional Cicho´n diagram. In fact, we can actually showthat a stronger fact is true. Theorem 1.3.25. All cuts consistent with the diagram are consistent with ZFC( W ) in thefollowing sense: Given any collection N of (not ∅ )-nodes in the diagram which are closedupwards under ⊆ there is a proper forcing P in W so that forcing with P over W results inall and only the nodes in N being nonempty. See Figure 1.12 for a pictorial representation Note that this is slightly weaker than the sense of cuts I have been considering abovesince I’m making no distinction between various non-empty nodes after forcing. Proof. There are two cuts I have yet to explicitly show. These correspond to e) and i) inFigure 1.12 below. However for completeness let me go through all cuts one at a time.a) All nodes are non empty: This is accomplished by LOC .b) All nodes except B ( ∈ ∗ ) are non empty: This is accomplished by D .c) All nodes below B ( ≤ ∗ ) are empty and D ( = ∗ ) is empty: This is accomplished by L . HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES ∅ B ( ∈ ∗ ) B ( ≤ ∗ ) B ( = ∗ ) D ( = ∗ ) D ( ≤ ∗ ) D ( ∈ ∗ ) ω ω \ ( ω ω ) W a) All nodes non empty ∅ B ( ∈ ∗ ) B ( ≤ ∗ ) B ( = ∗ ) D ( = ∗ ) D ( ≤ ∗ ) D ( ∈ ∗ ) ω ω \ ( ω ω ) W j) All nodes empty except ω ω \ ( ω ω ) W ∅ B ( ∈ ∗ ) B ( ≤ ∗ ) B ( = ∗ ) D ( = ∗ ) D ( ≤ ∗ ) D ( ∈ ∗ ) ω ω \ ( ω ω ) W i) All nodes below D ( ∈ ∗ ) empty ∅ B ( ∈ ∗ ) B ( ≤ ∗ ) B ( = ∗ ) D ( = ∗ ) D ( ≤ ∗ ) D ( ∈ ∗ ) ω ω \ ( ω ω ) W h) All nodes below D ( ∈ ∗ ) except B ( = ∗ ) are empty ∅ B ( ∈ ∗ ) B ( ≤ ∗ ) B ( = ∗ ) D ( = ∗ ) D ( ≤ ∗ ) D ( ∈ ∗ ) ω ω \ ( ω ω ) W f) All nodes below D ( ≤ ∗ ) empty and B ( = ∗ ) empty ∅ B ( ∈ ∗ ) B ( ≤ ∗ ) B ( = ∗ ) D ( = ∗ ) D ( ≤ ∗ ) D ( ∈ ∗ ) ω ω \ ( ω ω ) W e) All nodes below D ( ≤ ∗ ) empty and B ( = ∗ ) non empty ∅ B ( ∈ ∗ ) B ( ≤ ∗ ) B ( = ∗ ) D ( = ∗ ) D ( ≤ ∗ ) D ( ∈ ∗ ) ω ω \ ( ω ω ) W g) All nodes below D ( = ∗ ) empty and B ( = ∗ ) empty ∅ B ( ∈ ∗ ) B ( ≤ ∗ ) B ( = ∗ ) D ( = ∗ ) D ( ≤ ∗ ) D ( ∈ ∗ ) ω ω \ ( ω ω ) W d) All nodes below B ( = ∗ ) empty and D ( = ∗ ) non empty ∅ B ( ∈ ∗ ) B ( ≤ ∗ ) B ( = ∗ ) D ( = ∗ ) D ( ≤ ∗ ) D ( ∈ ∗ ) ω ω \ ( ω ω ) W c) All nodes below B ( ≤ ∗ ) empty, D ( = ∗ ) empty ∅ B ( ∈ ∗ ) B ( ≤ ∗ ) B ( = ∗ ) D ( = ∗ ) D ( ≤ ∗ ) D ( ∈ ∗ ) ω ω \ ( ω ω ) W b) All nodes except B ( ∈ ∗ ) non empty Figure 1.12: All Possible Cuts in the ≤ W Cicho´n Diagram. Each one can be achieved by aproper forcing over W . White means that the node is not empty while yellow means thatit is. No distinction is made between different non-empty nodes. Note that the trivial cutwhere all nodes remain empty is not shown. HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES B ( = ∗ ) are empty and D ( = ∗ ) is non empty: This is accomplished by E .e) All nodes below D ( ≤ ∗ ) are empty and B ( = ∗ ) is non empty: This is the first case where westill have to prove something. Let P = B ∗ ˙ PT . I claim that in W P this cut is realized. Wehave seen that forcing with B adds an eventually different real and, by further forcing with PT over W B will add a real which is unbounded by W B ∩ ω ω and hence W ∩ ω ω . It remainstherefore to see that in W P there are no dominating or infinitely often equal reals over W .To show that there are no dominating reals, note that in general PT adds no dominatingreal, so in W P there is no real which is dominating over W B . But, since B is ω ω -bounding,it follows that there is no real dominating over W in W P . To show there are no infinitelyoften equal reals, let us first note the following fact. Fact 1.3.26 (Corollary 2.5.2 of [7]) . Suppose M is a transitive model of a sufficiently largefragment of ZFC . Then M ∩ ω ∈ N if and only if there is a sequence h F n ⊆ n | n < ω i such that Σ ∞ n =0 | F n | − n < ∞ and for every x ∈ M ∩ ω there are infinitely many n so that x ↾ n ∈ F n . As a corollary of this Fact, notice that adding an infinitely often equal real on ω ω makesthe ground model reals measure 0. To see why, suppose y ∈ ω ω is infinitely often equal overan inner model M and let h τ k | k < ω i be an enumeration in M of the elements of 2 <ω . Thenfor every x ∈ ω ∩ M let ˆ x : ω → ω be defined by ˆ x ( n ) = k if x ↾ n = k . Clearly if x ∈ M the ˆ x ∈ M so there are infinitely many n such that ˆ x ( n ) = y ( n ). But then, pulling back,let y ′ : ω → <ω be defined by y ′ ( n ) = s k if g ( n ) = k and s k ∈ n and is trivial otherwise.Then we have that for every x ∈ M ∩ ω if ˆ x ( n ) = y ( n ) then x ↾ n = y ′ ( n ) so the sequence h{ y ′ ( n ) } | n < ω i witnesses that 2 ω ∩ M is measure 0 by the Fact.From this it follows immediately that P does not add infinitely often equal reals sinceboth B ([7, Lemma 6.3.12]) and PT ([7, Theorem 7.3.47]) preserve outer measure.f) All nodes below D ( ≤ ∗ ) are empty and B ( = ∗ ) is empty: This is accomplished by PT . HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES D ( = ∗ ) are empty and B ( = ∗ ) is empty: This is accomplished by C .h) All nodes below D ( ∈ ∗ ) except B ( = ∗ ) are empty: This is accomplished by B .i) All nodes below D ( ∈ ∗ ) are empty: This is the second cut where we still have something toprove. To achieve this one we force with the infinitely often equal forcing EE as defined in[7, Definition 7.4.11]. This forcing is ω ω -bounding so it doesn’t add reals to D ( ≤ ∗ ), does notmake the ground model reals meager (both of these facts are proved as part of [7, Lemma7.4.14]) so it doesn’t add reals to B ( = ∗ ) and generically adds a real which is infinitely oftenequal to all ground model elements of the product space Π n<ω n . Let’s see that EE addsa real to D ( ∈ ∗ ). Recall that this means there is a real which is not eventually captured byany ground model slalom. Let y : ω → <ω be the infinitely often equal real added by thegeneric and fix an enumeration h τ n | n < ω i (in W ) of 2 <ω . Let ˆ y : ω → ω be the functiondefined by ˆ y ( n ) = k if y ( n ) = τ k . I claim that this ˆ y is as needed. To see why, let s ∈ W bea slalom. We can associate (in W ) a function x s : ω → <ω by letting x s ( n ) be τ k where k is the least so that k / ∈ s ( n ) and τ k ∈ n . Note that such a k exists since | s ( n ) | = n . Since x s ∈ W there are infinitely many n so that x s ( n ) = y ( n ). Therefore there are infinitely many n so that ˆ y ( n ) / ∈ s ( n ), as needed.j) All nodes except ω ω \ ( ω ω ) W are empty: This is accomplished by S .k) All nodes are empty: This one is not pictured in Figure 1.12 since it is trivial. Let P beany forcing not adding reals, such as trivial forcing.To finish this section, let me observe one more analogue with the standard Cicho´n di-agram. Traditionally in the study of cardinal invariants of the continuum one sandwichesthe nodes in Cicho´n’s diagram on one side by ℵ , the smallest possible value of any node,and on the other side by 2 ℵ , the largest possible value of any node. One then views, for agiven model M of ZFC, the values of the other nodes on the diagram for M as a measure HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES M with regards to substantive, mathematical ap-plications. My diagram also naturally sandwiches itself between two invariants: the emptyset, the smallest possible value of any node, and the entirety of the new reals, ω ω \ ( ω ω ) W ,the largest possible value of any node. As such, I view my diagram studied in this paperas measuring, similar to the case of the Cicho´n diagram, the difference between the reals ofthe inner model W and the reals of V . A natural question to ask, therefore, is how strongthis “measurement” analogy is between these two diagrams. For example, in the genericextension of W by more than ℵ many Cohen reals, all nodes on the right side of the Cicho´ndiagram equal to 2 ℵ and all nodes on the left equal to ℵ , paralleling the situation I describedfor the model W [ c ]. However, in similar models studied for Hechler and eventually differentforcing, the nodes in the Cicho´n diagram still split into two cardinals, ℵ and ℵ , whereasthe diagram discussed in this paper automatically splits in three different sets of reals, asdiscussed. It appears that this may be necessary due to a result of Khomskii and Laguzzi instating that there is a canonical forcing in a certain sense to add infinitely-often-equal realsand this forcing does not add dominating reals, suggesting that perhaps there is no way thatboth B ( ≤ ∗ ) and D ( = ∗ ) can be nonempty and equal. ≤ W -Cicho´n Di-agram and the axiom CD ( ≤ W ) In this section building off the work done in the last section I build a model where there iscomplete separation between all elements in the diagram. Theorem 1.4.1. ( GBC ) Given any transitive inner model W of ZFC, there is a properforcing notion P , such that in W P all the (non- ∅ ) nodes in the ≤ W -Cicho´n diagram aredistinct and every possible separation is simultaneously realized. HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES ∅ B ( ∈ ∗ ) B ( ≤ ∗ ) B ( = ∗ ) D ( = ∗ ) D ( ≤ ∗ ) D ( ∈ ∗ ) ω ω \ ( ω ω ) W Figure 1.13: Full Separation of the ≤ W -diagramIn what follows I call the axiom “All consistent separations of the ≤ W -diagram are distinct” CD ( ≤ W ) or “full Cicho´n Diagram for ≤ W ”. Thus the above theorem states that CD ( ≤ W ) canbe forced over W by a proper forcing. For different inner models W the sentence CD ( ≤ W )may vary but they can all be forced the same way.Before proving this theorem I need a simple technical result about Sacks and Laverforcing. Lemma 1.4.2. The product forcing S × L satisfies Axiom A and hence is proper.Proof. Theorem 1 of [28] gives a general framework for showing that certain arboreal forcingnotions satisfy Axiom A (including Sacks and Laver forcings) and here I adapt the proof tothe case of a product of two arboreal forcing notions. Recall that if p, q ∈ S and n ∈ ω thenwe let q ≤ S n p if and only if q ⊆ p and every n th splitting node of q is an n th splitting node of p i.e. if τ ∈ q is a splitting node with n splitting predecessors in q then the same is true of τ in p . Also, given a canonical enumeration of ω <ω in which s appears before τ if s ⊆ τ and s ⌢ k appears before s ⌢ ( k + 1) then for p ∈ L one gets an enumeration of the elements of p above the stem, s p , ..., s pk , ... and if p, q ∈ L and n ∈ ω then let q ≤ L n p if and only if q ⊆ p and s pi = s qi for all i = 0 , ..., n . Clearly if for every n ∈ ω and ( p s , p l ) , ( q s , q l ) ∈ S × L we let( q s , q l ) ≤ n ( p s , p l ) if and only if q s ≤ S n p s and q l ≤ L n s l then this satisfies the first requirementof Axiom A forcings. Thus, it remains to show that for every S × L -name ˙ a and condition HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES p s , p l ) ∈ S × L if ( p s , p l ) (cid:13) ˙ a ∈ ˇ V then for every n there is a ( q s , q l ) and a countable set A ∈ V such that ( q s , q l ) (cid:13) ˙ a ∈ A .Fix such a name ˙ a and condition p = ( p s , p l ). Let D ⊆ S × L be the set of all ( q s , q l ) ≤ p such that there is some a ( q ) ∈ V with ( q s , q l ) (cid:13) ˇ a ( q ) = ˙ a . This set is dense below p since p forces ˙ a to be an element of V . Let H D ⊆ p be the set of all pairs ( s, τ ) ∈ p such thatthere is a ( s ′ , τ ′ ) ⊆ ( s, τ ) with s ′ n -splitting in p s and τ ′ n -splitting in p l and there is some r s,τ = ( r s , r l ) ≤ p in D whose stem (i.e. the pair of the stems from the two components)is ( s, τ ). Finally let M in ( H D ) be the set of ( s, τ ) ∈ H D which are minimal with respect toinclusion. Note that M in ( H D ) is an antichain since no two elements can be comparable andboth minimal. Let r = ( r s , r l ) = S { r s,τ | ( s, τ ) ∈ M in ( H D ) } . A routine check shows thatthe set r is a condition in S × L and r ≤ n p .Now let A = { a ( r τ,s ) | ( s, τ ) ∈ M in ( H D ) } . This set is countable thus to finish the lemmait suffices to show that r (cid:13) ˙ a ∈ ˇ A . To see this, suppose that t ≤ r and t (cid:13) ˙ a = ˇ a for some a . By extending t if necessary one may assume that the stem of t is in H D . But then someinitial segment of the stem is in M in ( H D ) so a ∈ A , as needed.Now I prove Theorem 1.4.1. Proof of Theorem 1.4.1. This essentially follows from the theorems of the previous section.Given a definable forcing notion Q let me write Q W for the version of that forcing notionas computed in W . Let P = S W × L W × LOC W . Then in W P not every new real is in anelement of the diagram since Sacks reals were added. Moreover, by our arguments above thecombination of LOC and L will add reals to every node of the diagram but, none of themwill be equal and moreover every possible non-separation is realized as one observes by myprevious arguments.It remains to see that P is proper. This follows from Lemma 1.4.2 plus the fact that LOC is σ -linked and hence indestructibly ccc. HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES CD ( ≤ W ). First, let me showthat there are other ways to obtain it. Indeed there is another, less finegrained approach toforcing CD ( ≤ W ). To describe this, let me make the following simple observation. Recall thatthe Maximality Principle MP of [30] states that any statement which is forceably necessaryor can be forced to be true in such as a way that it cannot become later forced to be false,is already true. If Γ is a class of forcing notions then the maximality principle for Γ, MP Γ ,states the same but only with respect to forcing notions in Γ. Proposition 1.4.3. The axiom CD ( ≤ W ) is forceably necessary, that is once it has beenforced to be true it will remain so in any further forcing extension. Thus in particular it isimplied by the maximality principle, MP .Proof. This is more or less immediate from the definition. Since CD ( ≤ W ) is defined relativeto a fixed inner model and the diagram for W concerns only the models W [ x ] for x ∈ ω ω ∩ V ,notice that forcing over V cannot change the theories of the models W [ x ] for x ∈ V hence if CD ( ≤ W ) is true in V it must remain so in any forcing extension. In other words absolutenessfor membership in each of the various classes holds and this guarentees that forcing cannotchange the relation x ∈ A for any node A of the diagram.Since CD ( ≤ W ) is forceably necessary it follows that M P implies CD ( ≤ W ).Now notice that since all the forcing notions used in Theorem 1.4.1 have size at most2 ℵ it follows that the collapse forcing Coll ( ω, < (2 ℵ ) + ) will add a generic making CD ( ≤ W )true. Since CD ( ≤ W ) is forceably necessary it follows that the full collapse forcing cannot killthe generic once it is added and, as a result one obtains Corollary 1.4.4. W Coll ( ω,< (2 ℵ ) + ) | = CD ( ≤ W )Moreover, note that while the forcing described in Theorem 1.4.1 was proper and hencepreserved ω the collapse forcing used above is not. Therefore the following is immediate. HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES Corollary 1.4.5. The statement “the reals of W are countable” is independent of the the-ory ZFC( W ) + CD ( ≤ W ) . Consequently CD ( ≤ W ) does not imply MP for any sufficientlydefinable W . Since CD ( ≤ W ) is forceably necessary and hence cannot be killed once it is forced tobe true it follows that any sentence which can be forced to be true from any model mustbe consistent with CD ( ≤ W ). Such examples include CH , 2 ℵ = κ for any κ of uncountablecofinality, Martin’s Axiom and its negation, ♦ and its negation, and a wide variety of forcingnotions associated with the classical Cicho´n’s diagram. In particular, CD ( ≤ W ) is independentof any consistent assignment of cardinals to the nodes in the Cicho´n diagram (cf [7] for avariety of examples of such).Let me finish now by showing the consistency of a strong version of CD ( ≤ W ), whichwas suggested to me by Gunter Fuchs. The idea is to iteratively force with the forcing P of Theorem 1.4.1 for long enough that a large collection of inner models W simultaneouslysatisfy CD ( ≤ W ). Theorem 1.4.6. Assume V = L . Then there is an ℵ -c.c. proper forcing extension inwhich ℵ = ℵ and for every ℵ -sized set of reals A there is an ℵ -sized set of reals B ⊇ A so that CD ( ≤ L [ B ] ) holds.Proof. Assume V = L and let ~ P = h ( P α , ˙ Q α ) | α < ω i be an ω -length countable supportiteration of copies of the forcing P from Theorem 1.4.1 (i.e. ˙ Q α +1 evaluates to ( P ) L P α ).Clearly ~ P is proper. Moreover, since CH holds in the ground model and the forcing P iseasily seen to be of size continuum, and does not kill CH it follows that ~ P has the ℵ -c.c.and every intermediate stage in the iteration preserves CH: L P α | = CH for all α < ω .However, since reals are added at every stage the final model satisfies 2 ℵ = ℵ .It remains to show that for every ℵ -sized set of reals A there is a set of reals B ⊇ A of size ℵ so that CD ( ≤ W ) holds for W = L [ B ]. Let A be a set of reals of size ℵ . Then, HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES α so that A ∈ L [ G α ] for G α be P α -generic. Note that we can code G α bya set of reals of size at most ℵ , say B , and without loss we can assume that A ⊆ B for L [ G α ] = L [ B ]. Then at stage P α +1 we added a generic witnessing that CD ( ≤ L [ B ] ) holds.Moreover, by the fact that this statement is forceably necessary, it cannot be killed by thetail end of the iteration so it holds in the final model.While it is not entirely clear what consequences we can expect from CD ( ≤ W ) for anarbitrary W , the stronger version obtained in Theorem 1.4.6 has several low hanging fruitsin this regard. Let me pluck a particularly simple one connecting the constructibility diagramto the standard Cicho´n diagram. Lemma 1.4.7. Assume for every ℵ -sized set of reals A there is an ℵ -sized set of reals B ⊇ A so that CD ( ≤ L [ B ] ) holds. Then all the cardinals in the Cicho´n diagram have size atleast ℵ .Proof. It suffices to show that add( N ) ≥ ℵ . Towards this goal, recall Bartoszy´nski’s char-acterization of add( N ) as the least cardinal κ so that there is a set of reals X of size κ sothat no single slalom can capture all the reals in X (Fact 1.1.2, part 1). The result is thenimmediate for, given any set of reals A of size ℵ , we can find a slalom s eventually capturingall reals in L [ B ] for some B ⊇ A so that CD ( ≤ L [ B ] ) holds so add( N ) > ℵ . I finish this chapter by collecting the open questions that have appeared. First I ask aboutthe Cicho´n diagram for other reduction concepts. Recall that in the case of ≤ T , the sets B ( ∈ ∗ ) and B ( ≤ ∗ ) were equal. Question . For which reductions ( ⊑ , x ) on the reals is B ⊑ ( ∈ ∗ ) ( B ⊑ ( ≤ ∗ )? HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES ⊑ equal to hyperarithmetic reduction, the equation B ⊑ ( ∈ ∗ ) = B ⊑ ( ≤ ∗ ) holds(hence it does for ≤ A as well). See [37, Fact 2.6]. This question has also been consideredin [37], see Problem 5.7 and the discussion preceding it. This shows that for many naturalreduction concepts the answer to the question above is negative. Taking this into account itseems reasonable to ask if indeed any “reasonable” reduction concept (whatever that means)provably does this in ZFC ? Note that if V = L all ≤ W relations are trivial.Next I ask about the ZFC( W )-provable relations between the nodes of the ≤ W -Cicho´ndiagram. While I have shown that there are no other implications it is entirely possible thatthere are other relations more generally. Question . What other ZFC( W )-provable relations are there between the sets in the ≤ W -diagram?My next collection of questions concerns the subforcings of LOC , a topic that deservesmore study. Question . What is the forcing adding the eventually different real described in Lemma1.3.24? Does it add a dominating real? Note that it must be ccc, in fact σ -linked and addeventually different reals which are bounded by nearly all ground model reals.Similarly, one might ask whether there is a similarly exotic subforcing of LOC for addinga dominating real. Question . Does every subforcing of LOC adding a dominating real add a D -generic real? Question . Does every subforcing of LOC add a Cohen real or a random real?There are also several open questions about the axiom CD ( ≤ W ). Question . What statements are implied by CD ( ≤ W )? In particular, does it imply thatthere are W -generics for the forcings to add reals we have discussed (Cohen, random, etc)? HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES Question . How does CD ( ≤ W ) relate to standard forcing axioms? In particular does MA ℵ imply CD ( ≤ L [ A ] ) for all ℵ -sized sets of reals A ? Does BPFA?Finally let me report on some ongoing attempts to generalize the work in this chapterfurther. After the results of this chapter were announced, J. Brendle pointed out to me thatthe setup described above can accommodate nodes explicitly corresponding to the cardinalcharacteristics for measure and category. Recall that for each real y , there is a canonicalBorel null (meager) set N y ( M y ) coded by y , see Lemmas 3.2 and 3.4 of [6]. For reals x and y , let x ∈ N y if and only if x ∈ N y and x ⊆ N y if and only if N x ⊆ N y and the same for M .These relations give the following bounding and non-dominating sets.1. B ( ⊆ N ) is the set of all reals x so that there is a y ≤ W x such that for each z ∈ W wehave that z ⊆ N y . In other words, in W [ x ] the union of all of the null sets coded in W is null.2. D ( ⊆ N ) is the set of all reals x so that there is a y ≤ W x such that for each z ∈ W wehave that y * N Z . In other words, there is a null set coded in W [ x ] which is not asubset of any null set coded in W .3. B ( ∈ N ) is the set of all reals x so that there is a y ≤ W x so that for each z ∈ W wehave that z ∈ N y . In other words, in W [ x ] the reals of W are measure zero.4. D ( ∈ N ) is the set of all reals x so that there is a y ≤ W x so that for each z ∈ Wy / ∈ N z . In other words in W [ x ] there is a real y not in any measure zero set coded in W . Note that x ∈ D ( ∈ N ) if and only if in W [ x ] there is a real y which is random over W i.e. y is B W generic. In the case of M , the corresponding statement is the samewith “random” replaced by “Cohen”.Working through the definitions and using the discussion from this chapter, it’s straight-forward to see that the following analogies hold: HAPTER 1. THE CICHO ´N DIAGRAM FOR DEGREES B ( ⊆ N ) ( B ( ⊆ M )) corresponds to add ( N ) ( add ( M ))2. D ( ⊆ N ) ( D ( ⊆ M )) corresponds to cof ( N ) ( cof ( M ))3. B ( ∈ N ) ( B ( ∈ M )) corresponds to non ( N ) ( non ( M ))4. D ( ∈ N ) ( D ( ∈ M )) corresponds to cov ( N ) ( cov ( M ))Moreover, using the theory of small sets worked out in [43] it’s not hard to show that theanalogous implications hold as well. Integrating these nodes with those from the diagramfor the combinatorial nodes gives one in which all of the equivalences hold from Facts 1.1.2and 1.1.3 with the exception of D ( = ∗ ) since as previously discussed Zapletal has shown thisone to be false. Thus we get the following diagram with 11 non-trivial nodes. B ( ⊆ N ) B ( ⊆ M ) B ( ∈ M ) D ( ∈ M ) D ( ⊆ M ) B ( ≤ ∗ ) D ( ≤ ∗ ) D ( ∈ N ) B ( ∈ N ) D ( ⊆ N ) B ( ∈ ∗ ) D ( ∈ ∗ ) ∅ B ( = ∗ ) ω ω \ W D ( = ∗ )Figure 1.14: Integrating the Combinatorial Nodes and the Nodes for Measure and CategoryThe following question is still open and seems related to several open problems concerningthe forcing Zapletal introduces in [51]. Question . Can the larger diagram including the nodes for measure and category be fullyseparated by proper (or at least ω -preserving) forcing in the same way that the diagramwith only the combinatorial nodes was in this chapter? hapter 2Cardinal Characteristics for Sets ofFunctions As we saw in the last chapter, many cardinal characteristics on ω ω arise as follows: fix somerelation R ⊆ ω × ω and let R ∗ ⊆ ω ω × ω ω be defined by x R ∗ y if and only if for all but finitelymany n we have x ( n ) R y ( n ). For instance, letting R be the the usual order on ω gives theeventual domination ordering. Each such R then gives rise to two cardinal characteristics, b ( R ∗ ), the least size of a set A ⊆ ω ω with no R ∗ -bound and d ( R ∗ ) the least size of a set D ⊆ ω ω which is R ∗ -dominating. A natural generalization of this is as follows: fix two sets X and Y , let I be an ideal on X and R ⊆ Y × Y be a binary relation on Y . Let Y X bethe set of functions f : X → Y and consider the relation R I ⊆ Y X × Y X given by f R I g if and only if for I -almost all x we have f ( x ) R g ( x ) i.e. { x ∈ X | ¬ f ( x ) R g ( x ) } ∈ I .Again we get two cardinal characteristics, this time on the set Y X : b ( R I ), the least size ofa set A ⊆ Y X which has no R I -bound and d ( R I ), the least size of a set D ⊆ Y X which isdominating with respect to R I . Note that letting X = Y = ω and I be the ideal of finitesets we recover the original setting for cardinal characteristics on Baire space and letting Y = 2 we recover the same for Cantor space.50 HAPTER 2. CARDINAL CHARACTERISTICS FOR SETS OF FUNCTIONS X = κ and Y = κ or 2 for arbitrary κ ,thus generalizing cardinal characteristics to larger cardinals, see for example the article [11]or the survey [36] for a list of open questions. In this case the interesting ideals are the idealof sets of size < κ , the non-stationary ideal, and, if κ has a large cardinal property, thenpotentially some ideal related to this. See [17], Theorems 6 and 8 for a particularly strikingresult relating cardinal invariants modulo different ideals.However, this framework is more flexible than just allowing one to study generalized Bairespace and Cantor space. Indeed it is easy to imagine numerous new cardinal characteristics.In this chapter I consider a different generalization, based on the function space ( ω ω ) ω ω offunctions f : ω ω → ω ω . Since Baire space comes with ideals that are not easily defined on κ κ we get further generalizations of cardinal characteristics. Specifically I will consider theideals N , M and K . The result is a “higher dimensional” version of several well-knowncardinal characteristics. While many different generalizations are possible let me stick withthe three relations we have already seen for simplicity: ≤ ∗ , = ∗ , and ∈ ∗ . By considering twocardinals for each of these three relations and three ideals I end up with 18 new cardinalscharacteristics above the continuum. The first main theorem of this chapter is to show thatthese “higher dimensional” cardinals behave, provably under ZFC , similar to their Bairespace analogues (the cardinals mentioned below will be defined in detail in the next section). Theorem 2.0.1. Interpreting → as ≤ the inequalities shown in Figures 2.1 and 2.2 are allprovable in ZFC . HAPTER 2. CARDINAL CHARACTERISTICS FOR SETS OF FUNCTIONS b ( ∈ ∗N ) b ( ≤ ∗N ) b ( = ∗N ) d ( = ∗N ) d ( ≤ ∗N ) d ( ∈ ∗N )Figure 2.1: Higher Dimensional Cardinal Characteristics Mod the Null Ideal b ( ∈ ∗M ) b ( ≤ ∗M ) b ( = ∗M ) d ( = ∗M ) d ( ≤ ∗M ) d ( ∈ ∗M ) b ( ∈ ∗K ) b ( ≤ ∗K ) b ( = ∗K ) d ( = ∗K ) d ( ≤ ∗K ) d ( ∈ ∗K )Figure 2.2: Higher Dimensional Cardinal Characteristics Mod the Meager and σ -CompactIdealsThe rest of this chapter is organized as follows. In the next section I introduce thecardinals b ( R I ) and d ( R I ) and basic relations between them are shown. The second sectioninvestigates the relation between these higher dimensional cardinal characteristics and thestandard cardinal characteristics on ω . Section 3 contains a number of consistency resultsand introduces three new forcing notions based on generalizations of Cohen, Hechler, andlocalization forcing. In section 4 I list a number of open questions, as well as some extensions. In this section I define the cardinals that will be studied for the rest of the chapter. Recallthat I write S for the space of slaloms. Definition 2.1.1. Let I ∈ {N , M , K} and R ∈ {≤ ∗ , = ∗ , ∈ ∗ } . HAPTER 2. CARDINAL CHARACTERISTICS FOR SETS OF FUNCTIONS b ( R I ) is the least size of a set A of functions from ω ω to ω ω for which there is no g : ω ω → ω ω ( g : ω ω → S in the case of R = ∈ ∗ ) such that for all f ∈ A the set { x ∈ ω ω | ¬ f ( x ) R g ( x ) } is in I .2. d ( R I ) is the least size of a set A of functions from ω ω to ω ω ( ω ω to S in the caseof R = ∈ ∗ ) so that for all g : ω ω → ω ω there is an f ∈ A for which the set { x ∈ ω ω | ¬ g ( x ) R f ( x ) } is in I .By varying I and R this definition gives 18 new cardinals. For readability, let me givethe details below for the case of the null ideal. Similar statements hold for M and K . Firstlet’s see explicitly what each relation R I is. On the two lists below let f, g : ω ω → ω ω and h : ω ω → S .1. f = ∗N g if and only if for all but a measure zero set of x ∈ ω ω we have that f ( x ) = ∗ g ( x ).2. f ≤ ∗N g if and only if for all but a measure zero set of x ∈ ω ω we have that f ( x ) ≤ ∗ g ( x ).3. f ∈ ∗N h if and only if for all but a measure zero set of x ∈ ω ω we have that f ( x ) ∈ ∗ h ( x ).For the cardinals now we get the following.1. b ( = ∗N ) is the least size of a = ∗N -unbounded set A ⊆ ( ω ω ) ω ω i.e. A is such that for each f : ω ω → ω ω there is a g ∈ A so that the set of { x | ∃ ∞ n g ( x )( n ) = f ( x )( n ) } is notmeasure zero.2. d ( = ∗N ) is the least size of a = ∗N -dominating set A ⊆ ( ω ω ) ω ω i.e. A is such that for every f : ω ω → ω ω there is a g ∈ A so that µ ( { x | f ( x ) = ∗ g ( x ) } ) = 1.3. b ( ≤ ∗N ) is the least size of a ≤ ∗N -unbounded set A ⊆ ( ω ω ) ω ω i.e. A is such that for each f : ω ω → ω ω there is a g ∈ A so that the set of { x | ∃ ∞ n f ( x )( n ) < g ( x )( n ) } is notmeasure zero. HAPTER 2. CARDINAL CHARACTERISTICS FOR SETS OF FUNCTIONS d ( ≤ ∗N ) is the least size of a ≤ ∗N -dominating set A ⊆ ( ω ω ) ω ω i.e. A is such that for every f : ω ω → ω ω there is a g ∈ A so that µ ( { x | f ( x ) ≤ ∗ g ( x ) } ) = 1.5. b ( ∈ ∗N ) is the least size of a ∈ ∗N -unbounded set A ⊆ ( ω ω ) ω ω i.e. A is such that for each f : ω ω → S there is a g ∈ A so that the set of { x | ∃ ∞ n g ( x )( n ) / ∈ f ( x )( n ) } is notmeasure zero.6. d ( ∈ ∗N ) is the least size of a ≤ ∗N -dominating set A ⊆ ( ω ω ) ω ω i.e. A is such that for every f : ω ω → ω ω there is a g ∈ A so that µ ( { x | f ( x ) ∈ ∗ g ( x ) } ) = 1.The first goal is to prove the following theorem, which shows that for each ideal the sixassociated cardinals fit together as in the case of the corresponding fragment of Cicho´n’sdiagram on ω , note not all cardinals in the ω case have analogues here. Theorem 2.1.2 (The Higher Dimensional Cicho´n diagram) . For an ideal I ∈ {N , M , K} and interpreting → as “is ZFC -provably less than or equal to” the following all hold: b ( ∈ ∗I ) b ( ≤ ∗I ) b ( = ∗I ) d ( = ∗I ) d ( ≤ ∗I ) d ( ∈ ∗I ) Proof. The proof of this theorem mirrors that of Theorem 1.2.7. Most of these implicationsare easy, however two are more substantial. The easy cases, exactly the same as thosefor Theorem 1.2.7 are shown below in Figure 2.3 and the arguments for these are exactlyidentical to those outlined in that proof. For instance, if A is ≤ ∗I -bounded, then of course itis not ≤ ∗I -dominating hence b ( ≤ ∗I ) ≤ d ( ≤ ∗I ). Similarly, if A ⊆ ( ω ω ) ω ω is a set so that there isa function h : ω ω → S so that for all f ∈ A f ∈ ∗I h then ˆ h ( x )( n ) = max h ( x )( n ) + 1 witnessesthe ≤ ∗I -bound of A . The other easy cases follow the same lines. HAPTER 2. CARDINAL CHARACTERISTICS FOR SETS OF FUNCTIONS b ( ∈ ∗I ) b ( ≤ ∗I ) b ( = ∗I ) d ( = ∗I ) d ( ≤ ∗I ) d ( ∈ ∗I )Figure 2.3: The Easy Cases of the Higher Cicho´n DiagramThe two more substantial inequalities are b ( ∈ ∗I ) ≤ d ( = ∗I ) and b ( = ∗I ) ≤ d ( ∈ ∗I ), so I turnmy attention to these. For the rest of this section, fix an ideal I ∈ {N , M , K} .The proofs of the inequalities consist of “lifting” the proofs for the Cicho´n diagram tothe higher dimensional case, particularly those in [5] or for Lemmas 1.2.8 and 1.2.9. Likein the proofs of those lemmas, fix finite, disjoint subsets of ω which collectively cover ω ,say J = { J n,k | k < n } . Let J n = S k I use the version of b ( ∈ ∗I ) defined in terms of J -slaloms as in the paragraph beforethe statement of the lemma. Let κ < b ( ∈ ∗I ). I need to show that κ < d ( = ∗I ). Fix a set A ⊆ ( ω ω ) ω ω of size κ . Let’s see that A is not = ∗I -dominating. To be clear, a set is = ∗I dominating if for every function f : ω ω → ω ω there is a g ∈ A so that for all x save for aset in I f ( x ) = ∗ g ( x ). Negating this, we need to find a function f : ω ω → ω ω so that for all HAPTER 2. CARDINAL CHARACTERISTICS FOR SETS OF FUNCTIONS g ∈ A the set { x | ∃ ∞ n g ( x )( n ) = f ( x )( n ) } is I -positive. In fact, I will show that under theassumption, such an f can be found so that each such set is I -measure one.Given an element of Baire Space, x : ω → ω let x ′ be the J -function defined by x ′ ( n ) = x ↾ J n . Note that since the J n ’s cover ω and are disjoint the function x x ′ is a bijection.Given a function f : ω ω → ω ω let f ′ similarly be defined by letting f ′ ( x ) = f ( x ) ′ . Let A ′ = { g ′ | g ∈ A } . Since this set has size κ it is ∈ ∗I -bounded i.e. there is a function f A with domain the set of J -functions and range the set of J -slaloms so that for all g ′ ∈ A ′ { x | g ′ ( x ) / ∈ ∗ f A ( x ) } ∈ I . I need to transform f A into a function f as advertized in theprevious paragraph. The crux of the argument is the following claim, which will also be usedin Lemma 2.1.5 below as well. Claim 2.1.4. Given a J -slalom s , there is a function x s : ω → ω so that for all y : ω → ω if y ′ ( n ) ∈ ∗ s ( n ) then there are infinitely many n < ω so that x s ( n ) = y ( n ) .Proof of Claim. Fix a J -slalom s . For each n let s ( n ) = { w n , ..., w nn − } . Define x s : ω → ω by letting for each n and k < n and l ∈ J n,k x s ( l ) = w nk ( l ). Suppose now that y : ω → ω is such that y ′ ( n ) ∈ s ( n ) for all but finitely many n < ω . Fix some n so that y ′ ( n ) ∈ s ( n ),say, y ′ ( n ) = w nk . Then for each l ∈ J n,k y ( l ) = w nk ( l ) = x s ( l ). Since there are cofinitely manysuch n ’s there are infinitely many such l ’s so x s is as needed.Now, returning to the proof of the lemma, let f : ω ω → ω ω be defined by letting f ( x ) bethe function x f A ( x ) in the terminology of the claim. In particular, if g : ω ω → ω ω then forevery x ∈ ω ω if g ′ ( x ) ∈ ∗ f A ( x ) then there are infinitely many n so that g ( x )( n ) = f ( x )( n ). Inparticular the set { x | g ′ ( x ) ∈ ∗ f A ( x ) } is contained in the set { x | ∃ ∞ n g ( x )( n ) = f ( x )( n ) } .For g ∈ A the former is I -measure one and so the latter is as well. As a result f is asneeded.By essentially dualizing the proof above we get as well the following. HAPTER 2. CARDINAL CHARACTERISTICS FOR SETS OF FUNCTIONS Lemma 2.1.5. b ( = ∗I ) ≤ d ( ∈ ∗I ) Proof. Suppose κ < b ( = ∗I ) and let A ⊆ ( S ) ω ω be of size κ . I need to show that there isan f ∈ ( ω ω ) ω ω so that for each h ∈ A the set of x so that f ( x ) / ∈ ∗ h ( x ) does not have I -measure one. For each h ∈ A let g h ∈ ( ω ω ) ω ω be defined by letting, for each x ∈ ω ω g h ( x ) = x h ( x ) as defined in the claim of the previous lemma. In particular, for each x notethat if f ( x ) ∈ ∗ h ( x ) then ∃ ∞ n g h ( x )( n ) = f ( x )( n ). Now let ¯ A = { g h | h ∈ A } . This sethas size at most κ so there is a = ∗I -bound by assumption, say f . This means that for each g h ∈ ¯ A we have that { x | g h ( x ) = ∗ f ( x ) } is I -measure one. But now the lemma is provedsince for every x so that g h ( x ) = ∗ f ( x ) by the contrapositive of the implication defining g h we have that f ( x ) / ∈ ∗ h ( x ).Combining the easy cases shown in Figure 2.3 with the proofs of the above two lemmasthen completes the proof of Theorem 2.1.2.Using the fact that every set in K is meager, we get the following relation between thediagrams for M and K . Proposition 2.1.6. The following inequalities are provable in ZFC : b ( ∈ ∗M ) b ( ≤ ∗M ) b ( = ∗M ) d ( = ∗M ) d ( ≤ ∗M ) d ( ∈ ∗M ) b ( ∈ ∗K ) b ( ≤ ∗K ) b ( = ∗K ) d ( = ∗K ) d ( ≤ ∗K ) d ( ∈ ∗K ) Proof. Fix a relation R . To see that b ( R K ) ≤ b ( R M ) note that if A ⊆ ( ω ω ) ω ω is R K -bounded, then it means that there is a function f : ω ω → ω ω so that for each g ∈ A the set { x | ¬ g ( x ) R f ( x ) } is ≤ ∗ -bounded by some z ∈ ω ω . But this means in particular that it ismeager and hence for each g ∈ A g R M f . HAPTER 2. CARDINAL CHARACTERISTICS FOR SETS OF FUNCTIONS d ( R M ) ≤ d ( R K ), suppose that A ⊆ ( ω ω ) ω ω is not R M -dominating. Thismeans that there is some f : ω ω → ω ω so that for each g ∈ A the set { x | f ( x ) R g ( x ) } isnot comeager. It follows that in particular it is not K -measure one then (since each such setis comeager) and therefore no g ∈ A is a R K bound on f , so A is not R K -dominating. This section concerns the relationship between provable inequalities between the cardinalsintroduced previously and cardinal characteristics of the continuum. I look first at therelationship between the higher dimensional cardinals and cardinal c + and then I comparethe diagrams for the null and meager ideals.I would like to argue that the standard diagonal arguments show that the cardinals definedabove are greater than or equal to c + , however this is not the case in ZFC alone. What istrue is that this holds under additional assumptions on certain cardinal characteristics on ω .For the statement of the lemma below, recall that non ( K ) = b , see Theorem 2.8 of [10]. Lemma 2.2.1. For each I ∈ {N , M , K} and R ∈ {∈ ∗ , ≤ ∗ , = ∗ } , if b ( R ) = non ( I ) = c then c + ≤ b ( R I ) . In particular, if add ( N ) = c then all 18 cardinals introduced in the previoussection are greater than c .Proof. This is essentially a generalization of the standard diagonal arguments used to showthat various cardinal characteristics are uncountable. The point is that in that case, therelations (on ω ) under consideration are always so that every finite set has an upper boundand the ideal is always the ideal of finite sets. It is exactly because arithmetic of cardinalcharacteristics is not so simple that the additional hypotheses are needed.Fix R and I and assume b ( R ) = non ( I ) = c . Let f α : ω ω → ω ω for each α < c . We want HAPTER 2. CARDINAL CHARACTERISTICS FOR SETS OF FUNCTIONS g : ω ω → ω ω (or g : ω ω → S in the case of R = ∈ ∗ ) so that for all α f α R I g . Thisis done as follows. First, list the elements of ω ω as { x α | α < c } . Next, note that for each β < c , by the fact that non ( I ) = c we have that { x α | α < β } ∈ I and, by the fact that b ( R ) = c we have that for each x γ ∈ ω ω the set { f α ( x γ ) | α < β } has an R -bound, say y γβ .Now define g so that g ( x α ) = y αα . It follows that for all α if γ > α then f α ( x γ ) R g ( x γ ) andsince the set { x γ | γ > α } is I -measure one we’re done.In ongoing joint work with J. Brendle we have since shown that the cardinals of the form d ( R I ) are provably at least c + but the cardinals of the form b ( R I ) can be both consistentlyequal to and strictly less than c , even ℵ with the continuum arbitrarily large. Hopefullythis will appear in print soon.Finally in this section let me compare the cardinals for M and N . Every argument givenso far has worked equally well for each of them, and the theorem below suggests that this isnot an accident. Theorem 2.2.2. If add ( N ) = cof ( N ) then for every relation R ∈ {∈ ∗ , ≤ ∗ , = ∗ } we havethat b ( R N ) = b ( R M ) and d ( R N ) = d ( R M ) . The proof of this theorem follows immediately from the following two lemmas, the firstof which is well known. Lemma 2.2.3 (Theorem 2.1.8 of [7]) . If add ( N ) = cof ( N ) then there is a bijection f : ω ω → ω ω so that for all A ⊆ ω ω f ( A ) ∈ N if and only if A ∈ M and f ( A ) ∈ M if and onlyif A ∈ N . Lemma 2.2.4. If there is a bijection f : ω ω → ω ω as in Lemma 2.2.3 then for every relation R ∈ {∈ ∗ , ≤ ∗ , = ∗ } we have that b ( R N ) = b ( R M ) and d ( R N ) = d ( R M ) .Proof. Fix a relation R ∈ {∈ ∗ , ≤ ∗ , = ∗ } and let f : ω ω → ω ω be a bijection as described inLemma 2.2.3. First, suppose that κ < b ( R N ) and let A ⊆ ( ω ω ) ω ω be a set of size κ . I claim HAPTER 2. CARDINAL CHARACTERISTICS FOR SETS OF FUNCTIONS g A : ω ω → ω ω ( g A : ω ω → S in the case R = ∈ ∗ ) so that for all g ∈ Ag R M g A and hence κ < b ( R M ). Let A f = { g ◦ f | g ∈ A } . Since f is a bijection | A f | = κ .By the hypothesis, let ¯ g : ω ω → ω ω be an R N bound on A f . I claim that g A = ¯ g ◦ f − isas needed. We have that for every g ∈ A if f − ( x ) = y is in the measure one set for which g ( f ( y )) R ¯ g ( y ) is true then the following holds: g ( x ) = g ( f ( f − ( x )) R ¯ g ( f − ( x )) = g A ( x )Therefore f − ( { x | ¬ g ( x ) R g A ( x ) } ) is contained in { x | ¬ g ( f ( x )) R ¯ g ( x ) } , which is nullby assumption and so the former is null as well. Hence by the property of f it follows that { x | ¬ g ( x ) R g A ( x ) } is meager so g A is an R M -bound as needed.This shows that b ( R N ) ≤ b ( R M ) however an identical argument, flipping the roles of themeager and null sets, shows the reverse inequality so we get that b ( R N ) = b ( R M ).An essentially dual argument works to show that d ( R N ) = d ( R M ). Let me sketch it,though I leave out the details. Assuming that κ < d ( R N ) we fix a set A ⊆ ( ω ω ) ω ω of size κ , define A f as before and let ¯ g be a function not dominated by any member of A f . Thenessentially the same argument shows that ¯ g ◦ f − is a function not dominated by any memberof A and, again by symmetry we obtain the required equality.Again in joint work with J. Brendle we have shown how to separate these. In particular,in the random model with c = κ we have shown b ( ∈ ∗N ) = ℵ < b ( ∈ ∗K ) = ℵ < b ( ∈ ∗M ) = κ + . In this section I consider consistent separations between the cardinals. For readability, Ifocus on the case of I = N , however, it’s routine to check that the arguments go through HAPTER 2. CARDINAL CHARACTERISTICS FOR SETS OF FUNCTIONS I = M and I = K . Indeed the essential point will be simply that N has a Borel baseand contains the countable subsets of ω ω . Also, I will only be considering models of CH soby Theorem 2.2.2 any separation between nodes in the N diagram will hold equally for the M diagram.From now on assume GCH holds and fix an enumeration of ω ω in order type ω , say { x α | α < ω } . Also fix an enumeration of the Borel sets in N in order type ω , say { N α | α < ω } . Suppose we have some forcing notion P which does not add reals. Note thatin this case if B is a Borel set then P forces that the name for B is equal to its evaluation inthe ground model. Also, since P does not add reals, it does not add any Borel sets either.This translates to the following idea, which is used in several proofs. Suppose ˙ A is a P namefor a subset of ω ω . If for some condition p ∈ P we have that p (cid:13) µ ( ˙ A ) = ˇ0, then we canalways find a q ≤ p and a Borel null set in the ground model N so that q (cid:13) ˙ A ⊆ ˇ N .The following simple lemma will be used in several proofs. Lemma 2.3.1. Let ~N = h N ,α | α < ω i and ~N = h N ,α | α < ω i be two sequences ofnull sets of length ω . There is an enumeration in order type ω , say h ( N ′ ,α , N ′ ,α ) | α < ω i of the set of all pairs ( N ,β , N ,γ ) so that for each α < ω we have x α / ∈ N ′ ,α ∪ N ′ ,α .Proof. First fix any enumeration of ~N × ~N , say h ( N ′′ ,α , N ′′ ,α ) | α < ω i and define inductivelyfor each α ( N ′ ,α , N ′ ,α ) to be the least γ so that ( N ′′ ,γ , N ′′ ,γ ) has not yet been enumeratedand x α / ∈ ( N ′′ ,γ , N ′′ ,γ ). I need to show that every ( N ′′ ,γ , N ′′ ,γ ) gets enumerated under thisprocedure. Suppose not and let γ be least so that ( N ′′ ,γ , N ′′ ,γ ) is not enumerated. Sincefor every β < γ the pair ( N ′′ ,β , N ′′ ,β ) was enumerated, there was some countable stageby which this happened and so for cocountably many α it must have been the case that x α ∈ N ′′ ,γ ∪ N ′′ ,γ . But this is impossible since N ′′ ,γ ∪ N ′′ ,γ is measure zero and hence cannotcontain a cocountable set. HAPTER 2. CARDINAL CHARACTERISTICS FOR SETS OF FUNCTIONS The point of this subsection is to prove the following theorem. Theorem 2.3.2 ( GCH ) . Let κ be a regular cardinal greater than ℵ . There is a cofinalitypreserving forcing notion P κ so that if G ⊆ P κ is V -generic then in V [ G ] we have c + = ℵ = b ( = ∗N ) < d ( = ∗N ) = 2 c = κ . The proof will involve an iteration of length κ of a certain forcing notion, C N . Let mebegin by introducing this forcing notion and studying its properties. Definition 2.3.3. The N -Cohen forcing, denoted C N , is the set of all p : dom( p ) ⊆ ω ω → ω ω so that dom( p ) and graph( p ) are both Borel and dom( p ) is measure zero. We let p ≤ q ifand only if p ⊇ q .The following observations are easy but will be useful. Proposition 2.3.4. The forcing C N is σ -closed and has size c , hence it has the c + -c.c. Inparticular, under CH all cofinalities and hence cardinalities are preserved.Proof. First let’s see that C N is σ -closed. Given a descending sequence p ≥ p ≥ p ... let p = S n<ω p n . Since the countable union of Borel sets is Borel it follows that p has a Borelgraph and since the countable union of null sets is null, it follows that p has null domain.Thus p is a condition so it is a lower bound on the sequence of p n ’s.To see that C N has size c it suffices to note that each condition is a Borel subset of ( ω ω ) ,of which there are only c many.Note that since C N adds no reals or Borel sets and every condition p ∈ C N is a Borel setit follows that (cid:13) C N ˙ C N = ˇ C N and so in particular, the product and iteration of C N are thesame. Now, a straightforward density argument shows that C N adds a function g : ω ω → ω ω ,namely the union of the generic filter. Indeed it’s easy to see that if p is any condition and HAPTER 2. CARDINAL CHARACTERISTICS FOR SETS OF FUNCTIONS N is any Borel null set then there is a condition q ≤ p so that N ⊆ dom( q ). I need to verifytwo properties of C N , given as Lemmas 2.3.5 and 2.3.6 below. The first will imply that inan iterated extension d ( = ∗N ) becomes large and the second will imply that b ( = ∗N ) remainssmall in an iterated extension. Lemma 2.3.5. If G ⊆ C N is generic over V then in V [ G ] the set of f ∈ ( ω ω ) ω ω ∩ V is notdominating with respect to the relation = ∗N .Proof. In fact a stronger statement is true, namely if g = S G and ˙ g is the name for g , thenfor any f : ω ω → ω ω in the ground model the set { x | f ( x ) = g ( x ) } is not measure zero.To see this, suppose that for some condition p and ground model function f we have that p (cid:13) µ ( { x | ˇ f ( x ) = ˙ g ( x ) } ) = ˇ0. Since every null set is contained in a Borel null set, there is aBorel Null set N , necessarily in the ground model since C N is σ -closed, and a strengthening q ≤ p so that q (cid:13) { x | ˇ f ( x ) = ˙ g ( x ) } ⊆ ˇ N . But now let x / ∈ N ∪ dom( q ) (this is possible since N ∪ dom( q ) ∈ N ). It is straightforward to verify that q ∗ = q ∪ {h x, f ( x ) i} is a conditionextending q but clearly q ∗ (cid:13) { x | ˇ f ( x ) = ˙ g ( x ) } * ˇ N , which is a contradiction. It follows inparticular that for every f ∈ V we have that on a non null set of x there are infinitely many n < ω so that f ( x )( n ) = g ( x )( n ). This implies the lemma. Lemma 2.3.6. If G ⊆ Π I C N is generic over V for the countable support product of C N over an index set I of size at most ℵ then in V [ G ] the set of f ∈ ( ω ω ) ω ω ∩ V is unboundedwith respect to the relation = ∗N .Proof. I need to show that in V [ G ] there is no h : ω ω → ω ω so that for all f : ω ω → ω ω in V the set of x for which f ( x ) = ∗ g ( x ) is measure one. Thus suppose for a contradiction thatthere is a condition p and a name ˙ h so that p (cid:13) ˙ h : ˇ ω ω → ˇ ω ω is such a function. I need todefine in V a function for which this fails.Note that (under CH ) Π I C N has size ℵ . For each condition p ∈ Π I C N , let N p be theunion of the domains of the coordinate conditions. Since Π I C N has countable support, it HAPTER 2. CARDINAL CHARACTERISTICS FOR SETS OF FUNCTIONS N p is null. Now, using Lemma 2.3.1 fix an enumeration h ( N ,α , p α ) | α < ω i of all pairs where N ,α ranges over the Borel null sets N α and p α is a condition in Π I C N ,and x α / ∈ N ,α ∪ N p α . For each α , let r α ≤ p α decide ˙ h ( x α ). Say that r α (cid:13) ˙ h (ˇ x α ) = ˇ y α for some y α . Let h ∗ : ω ω → ω ω be the function (defined in V ) so that h ∗ ( x α ) = y α forall α . Suppose that there is some Borel null set N and some condition p which forcesthat { x | ∃ ∞ n ˙ h ( x )( n ) = ˇ h ∗ ( x )( n ) } ⊆ ˇ N . Let α be such that ( N, p ) = ( N ,α , p α ). Then p α (cid:13) { x | ∃ ∞ n ˙ h ( x )( n ) = ˇ h ∗ ( x )( n ) } ⊆ ˇ N ,α . But r α ≤ p α forces that ˙ h ( x α ) = ˇ h ∗ ( x α ) and bythe choice of enumeration we had that x α / ∈ N ,α , which is a contradiction.I’m now ready to prove Theorem 2.3.2. In fact it follows from the following theorem,which is just a more precise statement of what will be shown. Theorem 2.3.7. Let κ be a regular cardinal greater than ℵ and let P κ be the countablesupport product of C N . Then P κ preserves cofinalities and cardinals and if G ⊆ P κ is V -generic then in V [ G ] c + = ℵ = b ( = ∗N ) < d ( = ∗N ) = 2 c = κ .Proof. Fix κ > ℵ regular, let P = P κ be the countable support product of C N of length κ .Clearly P is σ -closed and a straightforward ∆-system argument using GCH shows that it hasthe ℵ -c.c. It follows that all cardinals and cofinalities are preserved.Also, for each α the α -stage forcing P α adds a new function g α : ω ω → ω ω so in theextension 2 c ≥ κ . A standard nice name counting argument, again using GCH shows that infact 2 c = κ .It remains to show that ℵ = b ( = ∗N ) and d ( = ∗N ) = κ . For the first of these, it sufficesto see that ( ω ω ) ω ω ∩ V is unbounded with respect to = ∗N . To see this, by Lemma 2.3.6, itsuffices to note that if ˙ f is a name for a function in ( ω ω ) ω ω then ˙ f is equivalent to a Π I C N for I an index set of size ℵ . This latter statement is proved as follows: let, for each x ∈ ω ω A x be an antichain of conditions deciding ˙ f (ˇ x ) and note that the cardinality of the supportsof the elements of S x ∈ ω ω A x has size ℵ by CH using the countable support of the product. HAPTER 2. CARDINAL CHARACTERISTICS FOR SETS OF FUNCTIONS d ( = ∗N ) = κ , suppose that A ⊆ ( ω ω ) ω ω of size < κ . It follows that A musthave been added by some initial stage of the iteration, the next stage of which killed thepossibility that it was dominating by Lemma 2.3.5.Let me reiterate that, defining C M and C K in the obvious way the proofs can be repeatedverbatim to obtain similar consistencies for the M and K ideals. The same is true in theremaining subsections, though I won’t explicitly say this again. An interesting open questionthough is the following. Question . Are the forcing notions C N , C M and C K provably forcing equivalent? In this subsection I consider a generalization of Hechler forcing called D N and look at twomodels obtained by iterating this forcing. First I consider the countable support iterationof D N and then I look at a non-linear iteration of D N similar to the one used in [17]. In thelatter case I obtain the following consistency result. Theorem 2.3.8. Let ℵ ≤ κ ≤ λ with κ and λ regular. Then there is a forcing notion P κ,λ which preserves cardinals and cofinalities such that if G ⊆ P κ is generic then in V [ G ] wehave that b ( ≤ ∗N ) = κ < d ( ≤ ∗ ) = 2 c = λ . Similar to the last subsection I start by introducing the one step and studying its prop-erties. This forcing is reminiscent of Hechler Forcing. As before, I work with the null idealfor definiteness but it’s easy to see that the proofs adapt to the case of the other ideals. Definition 2.3.9. The N -Hechler forcing D N consists of the set of pairs ( p, F ) where p ∈ C N and F is a countable set of functions f : ω ω → ω ω . We let ( p, F ) ≤ ( q, G ) in case p ⊇ q , F ⊇ G and for all x ∈ dom( p ) \ dom( q ) and all g ∈ G g ( x ) ≤ ∗ p ( x ). HAPTER 2. CARDINAL CHARACTERISTICS FOR SETS OF FUNCTIONS d = ( p d , F d ) ∈ D N , let me call p d the stem of the condition and F d the side part . Thebasic properties I will need for D N are as follows. Proposition 2.3.10. D N is σ -closed and has the c + -c.c., thus assuming CH , it preservescofinalities and cardinals. Also, if G ⊆ D N is V -generic then the union of G is a function g : ω ω → ω ω so that for any f : ω ω → ω ω in the ground model the set of x so that g ( x ) doesnot eventually dominate f ( x ) is null.Proof. That D N is σ -closed is the same as the proof for C N . To see that it has the c + -c.c.it suffices to note that if two conditions have the same stem then they are compatible.Now to see that g is total is a simple density argument, noting that if d is some conditionand x / ∈ dom( p d ) then there is a y dominating all of the f ( x ) for f ∈ F d since this set iscountable and hence ( p d ∪ {h x, y i} , F d ) extends d as needed. Moreover, if f : ω ω → ω ω is afunction in the ground model and d is any condition then clearly we can strengthen d , sayto d ′ so that f is included in the side part d ′ . This strengthening forces, by the definition ofthe extension relation, that for all x / ∈ dom( p d ′ ) f ( x ) ≤ ∗ ˙ g ( x ). Since the domain of p d ′ wasmeasure zero this proves the second part. Remark . While it’s not used in any proof let me note that, unlike with C N it is not the casethat every condition in D N can be extended to include any Borel null set in the domain ofits stem. This is because given any uncountable Borel set, (at least under CH ) one can use asimple diagonal argument to build a function which is not dominated by any Borel functionon that set. What is true however, is that the stem of any condition can be extended toinclude any countable set.Let me now show what happens in the generic extension by a countable support iterationof D N . Theorem 2.3.11. Let κ be regular and let P κ be the countable support iteration of D N . If G ⊆ P κ is V -generic then in V [ G ] b ( ≤ ∗N ) = d ( = ∗ ) = κ = 2 c . HAPTER 2. CARDINAL CHARACTERISTICS FOR SETS OF FUNCTIONS Proof. Let κ > ℵ be regular and let P κ be the countable support iteration of length κ of D κ .That cardinals and cofinalities are preserved follows as for C N . Also, every set A ⊆ ( ω ω ) ω ω of size less than κ is added by some initial stage, after which a function bounding A wasadded so b ( ≤ ∗N ) = κ . Moreover a nice name argument easily gives that 2 c = κ .It remains to see that d ( = ∗N ) = κ in this model. For this, I use the fact that countablesupport iterations always add a generic for Add ( ω , 1) at limit stages of cofinality ω . Now,given any function f : ω → ω we can think of it as a function ˆ f from ω ω to ω ω by lettingˆ f ( x α ) = x β just in case f ( α ) = β . Suppose A ⊆ ( ω ω ) ω ω is a set of size less than κ . It musthave been added by some initial stage of the iteration P κ and therefore there is a later stagewhich adds an Add ( ω , g : ω → ω . By density, given any f : ω ω → ω ω in A and any Borel null set N we can find an x / ∈ N so that ˆ g ( x ) = f ( x ) and therefore, forany f ∈ A the set of x for which ˆ g ( x ) = f ( x ) is not null. Therefore in particular A is not a = ∗N -dominating family. Thus d ( = ∗N ) = κ .I’m now ready to prove Theorem 2.3.8. This uses a version of the iteration discussedin Section 3 of [17]. Let me recall the basics of what I need. Fix κ ≤ λ regular cardinalsgreater than or equal to ℵ and let Q = ( Q, ≤ Q ) be a well founded partial order so that b ( Q ) = κ and d ( Q ) = λ . For example, under GCH , κ × [ λ ] <κ ordered by ( α, τ ) ≤ ( β, σ ) ifand only if α < β and τ ⊆ σ is such an order, see Lemma 2 of [17] for a proof. I need todefine a σ -closed, ℵ -c.c. forcing notion D ( Q ) so that forcing with this partial order addsa cofinal embedding of Q into (( ω ω ) ω ω , ≤ ∗N ). If I can do this, then by Lemmas 3 and 5 of[17] it follows that in the extension by this forcing notion b ( ≤ ∗N ) = κ and d ( ≤ ∗N ) = λ . Forcompleteness, here are the cited lemmas. Lemma 2.3.12 (Lemma 3 of [17]) . If P and Q are partially ordered sets and P embedscofinally into Q then b ( P ) = b ( Q ) and d ( P ) = d ( Q ) . Lemma 2.3.13 (Lemma 5 of [17]) . Suppose P is a partial order with b ( P ) = β and d ( P ) = δ . HAPTER 2. CARDINAL CHARACTERISTICS FOR SETS OF FUNCTIONS