Aksjomatyzacja modelu Mathiasa w terminach gier

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1 Instytut Matematyczny Uniwersytetu Wrocławskiego rozprawa doktorska Aksjomatyzacja modelu Mathiasa w terminach gier Wojciech Stadnicki promotor: prof. dr hab. Janusz Pawlikowski Wrocław, 2014

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3 University of Wroclaw Mathematical Institute PhD thesis Axiomatization of the Mathias model in terms of games Wojciech Stadnicki supervisor: prof. Janusz Pawlikowski Wroclaw, 2014

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5 Streszczenie Przedmiotem badań zawartych w rozprawie jest jeden z najważniejszych modeli rozważanych w teorii mnogości, mianowicie model Mathiasa. Mimo że jest on otrzymany za pomocą iteracji długości ω 2 forcingu Mathiasa, w niniejszej rozprawie jego kombinatoryczne własności ujęto w języku deskryptywnej teorii mnogości. Wzorując się na aksjomacie CPA dla modelu Sacksa (zob. [7]), wprowadzono szereg aksjomatów, które coraz dokładniej odzwierciedlają kombinatoryczną strukturę rozpatrywanego modelu. Wykazano ich niesprzeczność oraz zbadano konsekwencje. W tym celu przeformułowano iterację forcingu Mathiasa, charakteryzując ją w terminach deskryptywnej teorii mnogości. Aksjomaty wyrażono przy użyciu zbiorów i funkcji borelowskich, ideałów na przestrzeniach polskich oraz gier i strategii. Opracowano w ten sposób aksjomatyzację modelu Mathiasa, która pozwala spojrzeć na jego strukturę od strony topologiczno-desktyptywnej, co znacznie ułatwia dalsze badania nad jego własnościami i prowadzi do nowych wyników. Dodatkowo uzyskano kilka wniosków dotyczących V -ultrafiltrów 1 indukowanych przez liczby rzeczywiste z modelu Mathiasa. Rozdział 1 wprowadza do tematyki rozprawy i nakreśla tło historyczne opisanych w niej badań. Zawiera skrócony opis poszczególnych rozdziałów, uwagi o możliwości zastosowania przedstawionych narzędzi do badania innych modeli, a także podziękowania. W Rodziale 2 zebrano wstępne, z reguły powszechnie znane fakty dotyczące rozważanych pojęć. Ma to w szczególności na celu ustalenie notacji oraz interpretacji symboli czy zwrotów. W Rozdziale 3 podano deskryptywną charakteryzację iterowanego forcingu Mathiasa, przy użyciu zbiorów borelowskich i ideałów na przestrzeniach polskich. Wykorzystano w tym celu metodę zastowaną w [7], opierając się na ideach z [21]. 1 filtrów maksymalnych na P(ω) V, gdzie V jest modelem wyjściowym v

6 W kolejnych rozdziałach wprowadzano poszczególne aksjomaty. W każdym przypadku wykazano, że rozpatrywany aksjomat jest spełniony w modelu Mathiasa, oraz rozważano jego konsekwencje. Rozdział 4 opisuje podstawowy aksjomat, CPA, analogiczny do przedstawionego w [7]. Implikuje on równość cov(j ) = x = ω 1, gdzie J oznacza σ-ideał zbiorów I kategorii Baire a lub zbiorów miary zero na prostej, natomiast x - dystrybutywność algebry r.o.(r, ) lub r.o.(c 0 \l 1, ) (zob. [9], [11]). Jego wzmocnienie, sformułowany w Rozdziale 5 aksjomat CPAs, dowodzi takich własności jak: h > ω 1, gdzie h oznacza dystrybutywność algebry (P(ω)/ fin), Hipoteza Borela (Borel Conjecture, zob. [2]), brak ultrafiltrów typu rapid na ω (zob. [18]), brak dalekich punktów rozspajających I U dla U ω (zob. [10]). Ten ostatni wynik jest nową własnością modelu Mathiasa, która dotychczas znana była dla modelu Lavera (zob. [10]). W Rozdziale 6 przedstawiono aksjomat SCPA, będący słabszą, mniej techniczną, taktyczną wersją aksjomatu SCPA (omówionego w Rozdziale 8). Jako wniosek z SCPA uzyskano równość h(2) = ω 1, gdzie h(2) oznacza dystrybutywność algebry (P(ω)/ fin) (P(ω)/ fin). Wynik Shelaha i Spinasa z [21], mówiący, że równość ta zachodzi w modelu Mathiasa, jest główną motywacją do poszukiwania silniejszych wersji aksjomatu CPA. Ponadto SCPA dowodzi, że dystrybutywność ((ω) ω, ) wynosi ω 1. ((ω) ω to rodzina nieskończonych partycji zbioru ω, z porządkiem X Y, gdy prawie wszystkie 2 elementy partycji X są sumami elementów partycji Y, zob. [22].) Rozdział 7 poświęcony jest aksjomatowi CPA, który oddaje strukturę modelu iteracyjnego związaną z aksjomatem, występującym w pośrednich rozszerzeniach o kofinalności ω 1. W Rozdziale 8 opisano zależności między rozważanymi dotychczas aksjomatami oraz podano ich uogólnienia, w szczególności wspomniany aksjomat SCPA, w jego pełnej, strategicznej wersji, a także aksjomat SCPAs, implikujący wszystkie poprzednie. Pozostałe rozdziały należy traktować jako dodatek do głównego tematu rozprawy. W Rozdziale 9 omówiono pewne własności V -ultrafiltrów w mo- 2 poza skończoną ilością vi

7 delu Mathiasa. W szczególności podano elementarne dowody uogólnień głównych lematów z [21]. (Rozumowania w [21] wykorzystują zaawansowane i nader techniczne narzędzia, co znacznie utrudnia ich śledzenie.) Rozdział 10 opisuje aksjomat mcpa modyfikację CPA, która implikuje kombinatoryczną zasadę. Z uwagi na techniczny charakter mcpa rozdział ten stanowi jedynie uzupełnienie opracowanej akjomatyzacji. vii

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9 Abstract The objective of this thesis is one of the most significant models considered in set theory, namely the Mathias model, which is cnstructed via the technique of iterated forcing. Although it is obtained by iteration of length ω 2 of Mathias forcing, we study its combinatorial structure using the framework of descriptive set theory. We present a series of axioms, modeled on the CPA axiom from [7], that describe the combinatorial core of the model. We prove their consistency and study their consequences. To this end, we give a descriptive set theoretical characterization of the iterated Mathias forcing. Our axioms are formulated in terms of Borel sets and functions, σ-ideals on Polish spaces, games and strategies. In this way we develope an axiomatization of the Mathias model, which gives a descriptive set theoretic insight into its structure, makes it more approachable, and leads to new results. As a byproduct, we obtain a few facts about V -ultrafilters 3 induced by reals from the generi extension via the iterated Mathias forcing. Chapter 1 is an itroduction to the topic, gives some background and history. It contains an overview of the thesis, remarks about possible application of developed methods for investigating other models, and acknowledgement. In Chapter 2 we gather the neccesary preliminaries as well as fix the notation and interpretation of symbols and phrases. In Chapter 3 we reformulate the iteration of Mathias forcing. This approach was considered in [7] and its adaptation to our case uses ideas of [21]. The following chapters describe the axioms. In each case, we prove that the considered axiom holds true in the model as well as discuss its consequences. In Chapter 4 we present the basic axiom, CPA, which is analogous to the one from [7]. It implies that cov(j ) = x = ω 1, where J is the σ-ideal of meager, or null sets, and x denotes the distributivity of Boolean alge- 3 maximal filters on P(ω) V, where V is the ground model ix

10 bras r.o.(r, ) or r.o.(c 0 \l 1, ) (see [9], [11]). Its modificatiom, the axiom CPAs, introduced in Chapter 5, proves such assertions as h > ω 1, where h is the distributivity of (P(ω)/ fin), Borel Conjecture (see [2]), the lack of rapid ultrafilters (see [18]), the lack of far cut points in I U for U ω (see [10]). The latter statement is a new property of the Mathias model, so far it was known for the Laver model. The motivation to search for a stronger version of CPA stems from the result of Shelah and Spinas from [21], which says that h(2) = ω 1 in the Mathias model. The axiom SCPA, which implies this equality, is formulated in Chapter 6. It is a weaker, tactic version of the axiom SCPA (introduced in Chapter 8) but its expression is much less technical. The axiom SCPA also proves that the distributivity of ((ω) ω, ) equals ω 1. ((ω) ω is the set of all infinite partitions of ω with the ordering X Y iff all but finitely many elements of X are unions of elements of Y, see [22].) Chapter 7 is devoted to the axiom CPA, which is a natural modifaction of CPA capturing some combinatorics provided by the principle, which holds in the intermediate generic extensions of cofinality ω 1. In Chapter 8 we discuss implications between the axioms introduced so far and formulate their generalizations. In particular, we present the full, strategic version of SCPA as well as the strongest axiom, called SCPAs, which implies all the previously stated. Two following chapters can be considered as an appendix to the main topic. In Chapter 9 we present some results concerning V -ultrafilters in the Mathias model. In particular, we give elementary proofs of two main Propositions from [21] and, in fact, we generalize them. (Original proofs are based on difficult and technically complicated methods, which make them hard to follow.) In Chapter 10 we modify CPA obtaining the axiom mcpa, which is strong enough to imply the principle. This modification is quite technical. We present it for for the sake of the completeness of the research. x

11 Contents Streszczenie (in Polish) Abstract Contents v ix xi 1 Introduction Background Overview Remarks Acknowledgement Preliminaries Distributivity Mathias forcing and Mathias model Iteration Iteration of countable length Two-step iteration Iteration of length ω Iteration of arbitrary length Axiom CPA Covering of meager and null ideals Distributivity of r.o.(r ) Distributivity of (c 0 \l 1, ) xi

12 5 Axiom CPAs Distributivity of P(ω)/ fin Borel Conjecture Sub-cut points in R Rapid ultrafilters Axiom SCPA Combinatorics of Mathias forcing Distributivity of (P(ω)/ fin) Distributivity of ((ω) ω, ) Axiom CPA 57 8 Generalizations Axiom CPAs Axiom SCPA Axiom SCPAs V -ultrafilters Axiom mcpa 73 Bibliography 77 xii

13 Chapter 1 Introduction The theory of proper forcing turned out to be one of the most important topics in modern set theory. Properness is a desirable attribute of forcing notions. Models obtained as generic extensions via the countable support iteration of proper forcing serve as the main source of consistency results concerning the cardinal characteristics of the continuum (see [5]). They play a significant role not only in set theory but also in the related areas. The Mathias model is an example of such an extension. It is obtain via the countable support iteration of length ω 2 of Mathias forcing. Its combinatorial properties proved to be useful in many fields of mathematics. Besides set theory one should mention Boolean algebras, Ramsey theory, measure theory and set-theoretic topology. Having the Laver property, Mathias forcing shares many features with Laver forcing. For instance, the Borel Conjecture, which says that every strong measure zero set is countable, holds in the Mathias model. Also, the iterated Mathias forcing increases many cardinal characteristics (they become ω 2 = c in the generic extension). On the other hand, the distributivity of some Boolean algebras remains small (equal to ω 1 ). Using the Mathias model, Shelah and Spinas obtained the consistency of h < h(2), where h is the distributivity of the algebra P(ω)/ fin and h(2) that of its square ([21]). This result is considered to be very significant. The main purpose of this thesis is to develope a comprehensive, descriptive set theoretic axiomatization of the Mathias model. This is done by investigating a series of axioms, which capture its combinatorial core. The axioms do not refere to the model itself and they are formulated in terms of games and strategies rather than forcing. In this way they shed a new light 1

14 2 CHAPTER 1. INTRODUCTION on the structure of the Mathias model and make it more approachable. 1.1 Background The concept of descriptive set theoretic axioms which reflect the combinatorial structure of some models obtained by the iterated proper forcing was started by Ciesielski and Pawlikowski. In [7] they introduced the Covering Property Axiom (CPA), which captures the combinatorial core of the Sacks model. Using CPA one can eliminate forcing and, instead of working with the iterated perfect set model, deduce directly from the axiom in the desctiptive set theoretic framework. Similar axioms can be stated for other models. They were rephrased in terms of ideals by Zapletal in [23]: if I is a nicely definable σ-ideal on an uncountable Polish space X, then the axiom CPA(I) holds in the model obtained by the countable support iteration of the forcing (Bor(X) \ I, ). Zapletal also proved a kind of completeness theorem: for a very tame cardinal invariant x, if x < cov(i) is true in some generic extension, then it must be a consequence of CPA(I). Zapletal s approach relies on the Fubini powers of the σ-ideal I. In order to have the necessary absoluteness, I must be iterable (see [23, Definition 5.1.3]). The assumptions on I are mild enough to catch the σ-ideals of meager, null, and countable sets, as well as the σ-ideal on the Baire space ω ω generated by σ-compact sets. These σ-ideals correspond to Cohen, radom, Sacks, and Miller forcings respectively. Mathias forcing correspondes to the σ-ideal of Ramsey null sets, which is not iterable, and Zapletal s approach fails. Indeed, according to the result of Sabok from [19], the set of codes for Ramsey positive analytic sets is Σ 1 2-complete. It follows that, without extra large cardinal assumptions, the countable support iteration of Mathias forcing can not be expressed using the Fubini powers of the σ-ideal of Ramsey null sets. Fortunatelly, an approach similar to that of [7] and using ideas of [21] works.

15 1.2. OVERVIEW Overview Throughout the thesis we introduce a series of axioms (CPA, SCPA, CPA, CPAs, SCPAs etc.). Although they are descriptive set theoretic statements, rather than forcing ones, they reflect combinatorial properties of the Mathias model. Each time we introduce an axiom we prove that it is true in the Mathias model and then study its consequences. The axioms are formulated in terms of games and strategies (or tactics, which are particular types of strategies). A typical application of an axiom needs a certain density lemma, which describes a strategy (or tactic) of one of the players. Usually, the lemma can be stated (and proved) in terms of forcing, as well as within the framework of descriptive set theory (both proofs would use the same argument expressed in different languages). Such a density lemma can always be considered as the black-box principle - its proof is irrelevant for the rest of the argument. The dissertation is organized as follows: Chapter 2 gathers some background and fixes the notation. The necessary preliminaries about the theory of forcing are given and the notion of distributivity of a Boolean algerba (or a partially ordered set) is introduced and characterized. In particular, the distributivity game is described. We will refere to it a number of times throughout the thesis, when calculating cardinal characteristics from our axioms. Finally, the Mathias forcing is introduced, together with a bundle of preliminary facts and the definition of the Mathias model. In Chapter 3 we present a descriptive set theoretic reformulation of the iterated Mathias forcing. This approach was developed by Shelah and Spinas in [21], and, in contrast to the one elaborated by Zapletal ([23]), it does not involve the Fubini powers. Anyway, this part is quite technical. Section 3.1 is devoted to the iteration of countable lenght. For α < ω 1, a σ-ideal I α on the Polish space ([ω] ω ) α is defined. Each I α is connected to the iteration of the Mathias forcing of lenght α. These σ-ideals are necessary to state our axioms. Section 3.2 describes in details the reformulation of the two-step iteration. Section 3.3 gives a glance at the limit step of the iteration. In Section 3.4 we give the full reformulation of the iteration of arbitrary lenght. This is needed to prove the consistency of our axioms. Therefore, the Reader who is interested just in applying the axioms, not in

16 4 CHAPTER 1. INTRODUCTION the consistency proofs, may simply skip this part after the end of Section 3.1 (Definition 3.10). The only exception is Theorem 10.3 from Chapter 10. Chapter 4 is dedicated to the Covering Property Axiom (CPA), which is the Mathias analoque of the axiom stated for the Sacks model in [7]. Although it is the weakest of the axioms considered in the thesis, it implies that x = ω 1, where x denotes the covering number of meager, or null sets, or the distributivity of Boolean algebras r.o.(r, ) or r.o.(c 0 \l 1, ) (see Sections 4.2, 4.3). In Chapter 5 we present CPAs, a modification of CPA, strong enough to imply such assertions as h > ω 1, Borel Conjecture, as well as the lack of rapid ultrafilters and far cut points in I U for U ω (see Section 5.3). The main motivation to search for a stronger version of the basic axiom CPA is the result of Shelah and Spinas from [21], which says that h(2) = ω 1 in the Mathias model. We want to formulate an axiom which implies this equality. This is done in Chapter 6, which is devoted to the tactic version of the Strong Covering Property Axiom (SCPA ). We also consider the result of Spinas from [22], which says that h(((ω) ω, )) = ω 1 in the Mathias model (see Section 6.3). Although the principle fails in the Mathias model, it is well known that it holds at the intermediate generic extensions of cofinality ω 1. In Chapter 7 we introduce CPA, which is a natural modifaction of CPA capturing some combinatorics provided by. In Chapter 8 we discuss implications between the axioms introduced so far and formulate their generalizations. In particular we present the full (strategic) version of the Strong Covering Property Axiom (SCPA). Moreover, we introduce the strongest axiom, called SCPAs, which implies all the previously stated. Two following chapters can be considered as an appendix to the main topic. Chapter 9 is devoted to the ground model ultrafilters on ω, induced by reals from the generic extension via the Mathias forcing (iterated with another notion with the Laver property). In particular, we give elementary proofs of Propositions 2.3, 2.4 from [21]. The main result of [21] follows from these Propositions quite easily. However, their original proofs are based on difficult and technically complicated methods, which make them hard to follow. In fact, we generalize these results using just the combinatorial

17 1.3. REMARKS 5 structure of conditions in the Mathias forcing. Finally, we discuss possible generalizations to the ground model ultrafilters added by P(ω)/ fin. In Chapter 10 we modify CPA obtaining the axiom mcpa, which is strong enough to imply the principle. This modification is quite technical. We present it for for the sake of completeness of the research. 1.3 Remarks We would like to point out that the axiomatization we developed for the Mathias model can be transfered to some other models obtained by the iteration of simply definable proper forcing. Indeed, it is not difficult to repeat the descriptive set theoretic reformulation of the iteration (Chapter 3) as well as to state the axioms and prove their consistency for Cohen, Sacks, Miller, Laver, or random forcings. It is natural to ask if any two of the axioms we introduce are equivalent. Although we heuristically conjecture the negative answer, we put no effort to find any models discerning them. The motivation was to catch the combinatorial core of the Mathias model by a series of descriptive set theoretic statements. Searching for models distinguishing them is a separate problem, though, and may be interesting in itself. 1.4 Acknowledgement I would like to thank Janusz Pawlikowski for all his support, care, and patience, so that this dissertation could ever come into being. All the years of his help and guidance, suggestions and ideas discussed at countless meetings, are much more than a supervision. Thank you, Janusz!

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19 Chapter 2 Preliminaries A forcing notion is a partially ordered set P, P with a maximal element. Stronger condition is the smaller one. Conditions p, q P are compatible iff there exists r P such that r P p and r P q. Otherwise they are incompatible, which is denoted by p P q or simply by p q. Let V be a countable transitive model of ZFC with P V (ZFC is a finite fragment of ZFC containing all the relevant axioms). A filter G on P is P-generic over V iff G D for every open dense D P such that D V. It is easy to show that for every p P there exists a P-generic filter over V which contains p (Rasiowa-Sikorski Lemma). If G is P-generic then V [G], the generic extension of the ground model V, is the minimal transitive model of ZFC such that V V [G] and G V [G]. For a P-name ẋ and a P-generic filter G over V the evaluation of ẋ with G in V [G] is denoted by ẋ/g. The P-generic filter G is considered as an object outside of the universe V. We refere to [20, I, 1] or [15, VII, 1] for a comprehensive discussion. Two forcing notions (P, P ) and (P, P ) are forcingwise equivalent iff for every P-generic filter G over V there is P -generic filter G V [G] such that V [G] = V [G ] and vice versa (see [20, I,Definition 5.2,p. 30]). The forcingwise equivalence is denoted by =. A function π: P P is a dense embedding if π preserves the ordering, i.e. p 0 P p 1 π(p 0 ) P π(p 1 ), π preserves the incompatibility, i.e. p 0 P p 1 π(p 0 ) P π(p 1 ), the image of π is dense, i.e. ( p P )( p P) π(p) P p. 7

20 8 CHAPTER 2. PRELIMINARIES If there exists a dense embedding π: P P then P = P. (see e.g. [15, VII, Theorem 7.11]). For an ordinal ε, by ε-iteration of a forcing notion we mean the countable support iteration of length ε. We refere to [14, Definition 16.29] for the general definition. The countable support means that we take inverse limits at limit steps of cofinality ω and direct limits at other limit steps. Equivalently, if P ε is an ε-iteration (and P ξ are names for iterands) and p P ε then for all but countably many ξ < ε the condition p ξ P ε ξ forces that p(ξ) is the maximal element of P ξ. The structure H κ consists of sets x such that the cardinality of the transitive closure of x is smaller than κ. Whenever H κ appears, we always assume that κ > ω 1 is regular and H κ contains everything relevant, in particular the forcing notion we consider, all its subsets etc. Let M H κ be a model of ZFC (not necessarily transitive). For a Borel set B we say that B M if the code #B of B (which can be seen as a real, the usual Π 1 1 coding is used, see [14, p. 504]) is in M. Whenever M is a fixed model and #B M, B M is the set computed in M from the code #B; we often drop the upperscript M if context permits. Note that if V [G] is a generic extension of V and B V then B V and B V [G] may be different sets (but B V = B V [G] V ). Similarily, if f is a Borel function, then by f M we mean that the code #f of f is in M. Moreover, when we work in M and #f M, then we often simply write f for the function f M computed in M from the code #f. If N H κ is countable (note that N is non-transitive) and P N, then we say that a filter G on P N is P-generic over N iff G D for any open dense set D P such that D N. As in [23], the expression N[G] denotes the generic extension of the transitive collapse N of N by the collapsed image G of G. If ẋ N is a P-name, then ẋ/g is the G evaluation of ẋ, the image of ẋ under the collapsing function. If ẋ N is a P-name for an element of ω ω then the set A = {y ω ω : ( G P M) G is P-generic over N & y = ẋ/g} is Borel (see [23, Fact 1.4.8]). Moreover, the space G of all filters on P N which are P-generic over N is Polish 1 and the function e: G A given by 1 with the topology generated by the family {G p : p P N} as the base for closed sets, where G p = {G G: p G}

21 2.1. DISTRIBUTIVITY 9 e(g) = ẋ/g is a Borel bijection. In particular, for every y A the P-generic filter G over N such that ẋ/g = y is unique. If G is a filter on P (not necessarily contained in N), then by saying that G is P-generic over N we mean that G N is P-generic over N. We simply write N[G] and ẋ/g in place of N[G N] and ẋ/(g N). A forcing notion P is proper if for each countable N H κ such that P N and every condition p P N there exists q P p which is N-master, i.e. for every open dense D P such that D N we have q Ġ Ď Ň. Properness is preserved under countable support iteration (see [20]). If P is proper then P preserves ω 1, i.e. ω V [G] 1 = ω1 V whenever G is P-generic over V. A forcing notion P has the Laver property if for any p P, any P-name ġ for an element of ω ω, and any collection S(n): n ω V such that ( n ω) S(n) 2 n and p ( n ω) ġ(n) S(n) there exists q P p and T (n): n ω V such that ( n ω) (T (n) S(n) & T (n) n) and q ( n ω) ġ(n) T (n). The Laver property is preserved under countable support iteration (see [2]). 2.1 Distributivity Recall that a complete Boolean algebra B is κ-distributive if u α,i = u α,f(α) α<κ i I α α<κ f α<κ Iα for any u α,i : α < κ, i I α B (see [14, p. 85]). For any partially ordered set P there is a unique complete Boolean algebra r.o.(p) such that the separative quotient of P can be densely embedded into r.o.(p). The distributivity h(p) of P is defined as the minimal κ such that r.o.(p) is not κ-distributive. The symbol h denotes the distributivity of the algebra P(ω)/ fin and h(2) is the distributivity of its square (P(ω)/ fin) (P(ω)/ fin). We introduce the distributivity game G(P, α) to characterize h(p) (see [21, Introduction]).

22 10 CHAPTER 2. PRELIMINARIES Definition 2.1. Let P be a partial order and let α be an ordinal. Consider the game G(P, α) between two players PI and PII. The game lasts α rounds. PII begins with some initial p in P. Then PI chooses some p I 0 P p in, PII chooses p II 0 P p I 0, and so on. At step γ, PI plays (if possible) p I γ P which is below p II β for every β < γ. Then PII responds with pii γ P p I γ. PI wins iff he can always pick a legal p I γ during the play and there is a p P such that p p II β for all β < α. If r.o.(p) is homogenous, the initial move p in of PII may be skipped. Fact 2.2 ([21]). Suppose that P is a partial order such that r.o.(p) is homogenous. The following conditions are equivalent: 1. P is κ-distributive. 2. The intersection of any family {D γ : γ < κ} consisting of open dense sets 2 in P is dense. 3. The intersection of any decreasing family {D γ : γ < κ} consisting of open dense sets 3 in P is dense. 4. PII has no winning strategy in the distributivity game G(P, κ). 2.2 Mathias forcing and Mathias model Denote the sets of finite and infinite subsets of ω by [ω] <ω and [ω] ω respectively. We consider [ω] ω as a topological subspace of the Cantor set 2 ω. Recall the Mathias forcing with the ordering Q = { s, A : s [ω] <ω, A [ω] ω & max(s) < min(a)} s, A Q t, B t s t B & A B. It is well known that Q is proper and has the Laver property. The topology on Q is inherited from [ω] <ω [ω] ω, where [ω] <ω is discrete. For q Q let 2 If P is not separative then we assume that each D γ is closed under the separation equivalence: i.e. if x D γ and y P satisfies ( z P)(z x z y), then y D γ. 3 As above.

23 2.2. MATHIAS FORCING AND MATHIAS MODEL 11 s q [ω] <ω and A q [ω] ω denote its finite and infinite part respectively, i.e. q = s q, A q. Let G be Q-generic filter over V. The generic real associated with G is given by the formula r = {s [ω] <ω : ( A [ω] ω V ) s, A G}. Note that V [r] = V [G], since G = { s, A Q V : s r s A}. Moreover, any real x [ω] ω induces a filter G x = { s, A Q: s x s A} on Q. We say that x is Mathias over N if G x is Q-generic over N. The following result is due to Mathias. Theorem 2.3 ([14, Corollary 26.38]). If x [ω] ω is a Mathias real over N and y x is infinite, then y is Mathias over N. We shall fix some further notation. If s [ω] <ω and n > max(s) (we agree on min( ) = max( ) = 0) then s n = s {n}. Moreover, if s, t [ω] <ω are such that max(s) < min(t), or one of them is empty, then s t = s t. If A [ω] ω and n ω then by A/n we mean A \ (n + 1). Recall the following well-known lemma: Lemma 2.4 ([2, Lemma 7.4.6]). Suppose that s, A Q and ψ is a sentence in the forcing language. There exists an infinite B A such that either s, B Q ψ or s, B Q ψ. Let Q ε be the ε-iteration of Q. We will identify Q 1 with Q (note that formally Q 1 = { q : q Q}). Let Q 0 denote the trivial poset { }. For any ε the notion Q ε is proper and has the Laver property. Recall that the Mathias model is obtained as a Q ω2 -generic extension V [G] of V. Fix ε and ξ < ε. Let G ε denote a Q ε -generic filter over V. Then G ξ = G ε ξ = {p ξ: p G ε } is Q ξ -generic over V. By ṙ ξ we denote the cannonical Q ξ+1 -name for the ξ-th generic real and by r ξ its evaluation in the generic extension V [G ξ+1 ] V [G ε ]. For x [ω] ω let x ω ω be the increasing ennumeration of x. We use ṙ ξ to denote the cannonical Q ξ+1 -name for r ξ. For p Q ε let cl(p) ε be any countable set of ordinals such that p depends only on ṙ ζ : ζ cl(p) (for example, find a countable N H κ, where ε, p N and take cl(p) = N ε). 4 4 Note that cl(p) is not necessarily equal to the support of p, but it must contain it.

24 12 CHAPTER 2. PRELIMINARIES Sometimes we will deal with forcing notions of the form P = Q P, where P has the Laver property. Then r, ṙ, r and ṙ denote the generic real added by Q, its cannonical name, its ennumeration, and the cannonical name for the ennumeration respectively.

25 Chapter 3 Iteration The reformulation of iterated forcing in terms of ideals and descriptive set theory developed by Zapletal (see [23, Chapter 5]) does not catch Mathias forcing unless large cardinals are used to guarantee enough absoluteness. Fortunatelly, a ZFC-only approach, similar to that of [7], is possible using ideas of Shelah and Spinas from [21]. We present it in this chapter. For any ordinal ε, we will define a forcing notion P ε = Qε that provides the necessary absoluteness. This is quite technical and we will often cite [21]. However, we sketch enough details to give, hopefully, the flavor of the method. In particular, we present a detailed proof that P 2 = Q2 (Section 3.2). First we will deal with iterations of countable length. 3.1 Iteration of countable length Definition 3.1. For α < ω 1 we put p P α if and only if: (i) p is a function and dom(p) = α, (ii) for β < α, p(β) is a code for a Borel function, p(β): ([ω] ω ) β Q. The idea of Definition 3.1 is that p(β) assignes in a Borel way a Mathias condition to any sequence of reals of length β. To define Pα we first assign a Borel set B p ([ω] ω ) α to each p P α. The definition proceeds by induction on α: α = 0: P 0 = { } and B = ([ω] ω ) 0 = { }. 13

26 14 CHAPTER 3. ITERATION α = β +1: x γ : γ < α B p if x γ : γ < β B p β (note that p β P β ) and s q x β s q A q, where q = p(β) ( x γ : γ < β ), (i.e. q is in the filter on Q induced by x β ). α = α: x γ : γ < α B p if for all β < α we have x γ : γ < β B p β. A sequence from B p is said to satisfy p. For p, p P α we define p Pα p B p B p. One can easily show that p Pα p iff for every x γ : γ < α B p and for each β < α p(β) ( x γ : γ < β ) Q p (β) ( x γ : γ < β ). Theorem 3.2 (cf. [21, Corollary 3.7]). For any α < ω 1 we have P α = Qα. The complexity of (P α, Pα ) is Π 1 1 so its definition is absolute between transitive models of large enough finite fragment of ZFC. Therefore, we have an absolute description of the iteration of Mathias forcing of countable length. The proof of Theorem 3.2 boils down to showing a dense embedding of P α into Q α ([15, VII, Definition 7.7]). The following definition and lemma play the key role. Definition 3.3. Define the function π: P α Q α as follows: π(p) = q, where q(0) = p(0)( ), q(γ) is a Q γ -name such that Q γ q(γ) = p(γ)( ṙ β : β < γ ). Lemma 3.4 ([21, Lemma 3.8]). Let α < ω 1. Suppose that N H κ is countable, α N, and p P α N. There exists p Pα p such that every sequence x B p is Q α -generic over N, i.e. there is a (necessarily unique) filter G x Q α N which is Q α -generic over N, and x = ṙ γ : γ < α /G x = ṙ γ /G x : γ < α. If G α is Q α -generic, then {p P α : π(p) G α } is P α -generic. We will also denote it with G α (and its cannonical P α -name with Ġα). It should make no confusion. An inductive argument gives the following fact (c.f. Fact 3.26).

27 3.1. ITERATION OF COUNTABLE LENGTH 15 Fact 3.5. Q α (π(p) Ġα ṙ γ : γ < α B p ), for every p P α. It allows us to treat ṙ γ as a P α -name (in fact P γ+1 -name) in the following way: ṙ γ : γ < α is the P α -name for the unique sequence in ([ω] ω ) α, which satisfies every p Ġα. Moreover, for every p P α, ) P α (p Ġα ṙ γ : γ < α B p. Fact 3.6. Let B ([ω] ω ) α be a Borel set. For every p P α there exists p Pα p such that either B p B or B p B =. Proof. There exists p Pα p such that either p Pα ṙ γ : γ < α B or p Pα ṙ γ : γ < α B c. Without lost of generality assume the former. Let N H κ be a countable structure, containing α, B and p. By Lemma 3.4 there is p Pα p such that for every sequence x B p the filter G x(ξ) : ξ < α is Q α -generic over N. Fix such x B p. By genericity N[ x] = x B and by absoluteness x B. It follows that B p B. Definition 3.7. For each α < ω 1 let I α be the σ-ideal of subsets of ([ω] ω ) α generated by Borel sets B such that ( p P α )( p Pα p) B p B =. Remark 3.8. For Borel sets being I α -positive is a Σ 1 2 statement. As a corollary from Facts 3.5 and 3.6 we get: Corollary 3.9. For a Borel set B ([ω] ω ) α we have: (i) B / I α ( p P α ) B p B (ii) B I α P α ṙ γ : γ < α / B ( p P α )( p Pα p)b p B =. Definition Define Q α = Bor(([ω] ω ) α ) \ I α. Arguing as in the proof of Fact 3.6 one can show that the function ι: (P α, Pα ) (Q α, ) given by ι(p) = B p is a dense embedding. Hence, the ordering (Q α, ) is forcingwise equivalent to Q α. We will often treat ṙ γ as the Q α-name (Q γ+1-name): the γ-th coordinate of the unique member in the intersection of conditions in Q α-generic filter.

28 16 CHAPTER 3. ITERATION 3.2 Two-step iteration In this section we consider the two-step iteration and give a proof Theorem 3.2 when α = 2. The complete proof in [21] is rather technical, the two step case gives a good insight into its ideas. According to our notation Q 2 is the classical two-step iteration of the Mathias forcing. For q Q 2 and i = 0, 1 let us denote q i = q(i) (formally q is a sequence of length 2). So, for q 0, q 1, q 0, q 1 in Q 2 we have q 0, q 1 Q2 q 0, q 1 q 0 Q q 0 & q 0 Q q 1 Q q 1. Recall that p P 2 iff p = p(0), p(1), where p(i): ([ω] ω ) i Q is a Borel function for i = 0, 1. Let us denote p(i) by p i. Note that formally p 0 : { } Q. However, it seems ridiculous to keep this notation, so we will treat p 0 simply as a condition from Q, writing p 0 rather than p 0 ( ). Hence, p = p 0, p 1 P 2 iff p 0 Q and p 1 : [ω] ω Q is a Borel function. We associate with p = q Q = P the Borel (in fact closed) set B p = {x [ω] ω : s q x s q A q }, and with p = p 0, p 1 P 2 the Borel set B p = { x 0, x 1 ([ω] ω ) 2 : x 0 B p0 & s p 1(r 0 ) x 1 s p 1(r 0 ) A p 1(r 0 ) }. For p, p P 2 we have p P2 p B p B p p 0 Q p 0 & ( x B p0 ) p 1 (x) Q p 1(x). There is the cannonical function π: (P 2, P2 ) (Q 2, Q2 ), given by π(p) = p 0, p 1 (ṙ 0 ), where p = p 0, p 1 and ṙ 0 is the cannonical name for the Mathias real, added by the first iterand. (By p 1 (ṙ 0 ) we mean any Q-name q such that Q q = p 1 (ṙ 0 ).) Note that π is not one-to-one. For p = p 0, p 1, p = p 0, p 1 in P 2 we have π(p) Q2 π(p ) p 0 Q p 0 & p 0 Q p 1 (ṙ 0 ) Q p 1(ṙ 0 ). Fact Let p = p 0, p 1 P 2. Then Q 2 π(p) Ġ2 ṙ 0, ṙ 1 B p.

29 3.2. TWO-STEP ITERATION 17 Proof. Suppose that q = q 0, q 1 Q 2 and q Q2 π(p) Ġ2. Then q 0 Q p 0 & q 0 Q q 1 Q p 1 (ṙ 0 ). Therefore, q 0 Q ṙ 0 B p0 and q Q2 s p 1(ṙ 0 ) ṙ 1 A p 1(ṙ 0 ). It follows that q Q2 ṙ 0, ṙ 1 B p. On the other hand, suppose that the condition q = q 0, q 1 Q 2 forces that ṙ 0, ṙ 1 B p. Then q Q2 (ṙ0 B p0 s p 1(ṙ 0 ) ṙ 1 A p 1(ṙ 0 ) ). Since ṙ 0 is a Q-name, we have q 0 Q ṙ 0 B p0. But it means that q 0 Q p 0. Moreover, q 0 Q q 1 Q p 1 (ṙ 0 ). Hence, q Q2 π(p), so q Q2 π(p) Ġ2. Corollary The function π is order preserving, i.e. for every p, p P 2, if p P2 p then π(p) Q2 π(p ). Proof. Suppose that p P2 p. By Fact 3.11, π(p) Q2 ṙ 0, ṙ 1 B p. Since B p B p, we have π(p) Q2 ṙ 0, ṙ 1 B p. Using the fact again we get π(p) Q2 π(p ) Ġ2, hence π(p) Q2 π(p ). Observe that the converse is not true, i.e. there exist p, p P 2 such that π(p) Q2 π(p ) holds, but p P2 p fails (take any p, p such that p 0 Q p 0 and the set {x B p0 : p 1 (x) Q p 1(x)} is non-empty and Ramsey null). Lemma The image of π is dense in Q 2, i.e. for each q = q 0, q 1 Q 2 there exists p = p 0, p 1 P 2 such that π(p) Q2 q. Proof. Let N be a countable elementary submodel of H κ with q N. Take p 0 = q 0. If x is Mathias over N such that q 0 G x then define p 1 (x) = q 1 /G x, i.e. the evaluation of q 1 in the model N[G x ]. Otherwise let p 1 (x) =, ω. It is easy to verify that π(p) Q2 q. Lemma Suppose that p, p P 2 and π(p) Q2 π(p ). There exists p P 2 such that p P2 p and p P2 p. Proof. Let N H κ with p, p N. The following Claim is a special case of Lemma 3.4, where α = 1. Claim There is p 0 below p 0 such that every x B p 0 N. In particular p is an N-master condition. is Mathias over

30 18 CHAPTER 3. ITERATION Proof. Let y be Mathias over N, such that p 0 G x. Define p 0 = s p 0, y \s p 0. Then every x from B p is a subset of y, so by Theorem 2.3 it is Mathias 0 over N. Now let p 1 = p 1. Define p = p 0, p 1. Clearly p P2 p. We claim that p Q2 p. Obviously p 0 Q p 0. Fix x B p. The filter G 0 x is Q-generic over N. Since p 0 Q p 0 Q p 1 (ṙ 0 ) Q p 1(ṙ 0 ) we have N[G x ] = p 1(x) = p 1 (x) Q p 1(x) (the evaluation of ṙ 0 in N[G x ] is x). By the absoluteness of Q we obtain p 1(x) Q p 1(x). Since x B p was arbitrary, we have p 0 P2 p. Corollary (P 2, P2 ) = (Q 2, Q2 ). Proof. By Corollary 3.12 and Lemma 3.13 π is order preserving and its image is dense in (Q 2, Q2 ). By Lemmas 3.14 and 3.13, π preserves incompatibility. Therefore, π is a dense embedding. We end this section proving Lemma 3.4 for α = 2. We would need it to show that P 3 = Q3. Lemma Suppose that N H κ is countable and p = p 0, p 1 P 2 N. There exists p = p 0, p 1 P2 p such every x 0, x 1 B p is Q 2 -generic over N, i.e. x 0 is Mathias over N and x 1 is Mathias over N[G x0 ]. Proof. Use Claim 3.15 to find p 0 Q p 0 such that each real in B p 0 is Mathias over N. It remains to define the Borel function p 1: [ω] ω Q. If x / B p 0 let p 1(x) = p 1 (x). For x B p 0 define p 1(x) in the following way: Let Ḋ ξ : ξ < ν N be a Q-name for the sequence of all open dense subsets of Q. Let Ḋn: n ω V be an ennumeration of {Ḋ ξ : ξ ν N}. Then for any x B p 0 the sequence Ḋn/G x : n ω V lists all dense subsets of Q from the model N[G x ]. Now for each n ω let q ζ n: ζ < µ N be a Q-name for a sequence of all elements of Ḋ n. For x B p 0 we have a matrix q ζ n/g x: n ω, ζ µ N such that its n-th row ennumerates the set (Ḋn/G x ) N[G x ]. The assignment x ġζ n/g x is Borel for each n ω and ζ µ N.

31 3.3. ITERATION OF LENGTH ω 19 We will treat the countable well-ordered set A = µ N as an alphabet and for each x B p 0 we will construct a subtree T x of A <ω. Fix x B p 0. The tree T x is obtain by the following procedure: the empty sequence belongs to T x, ζ 0 T x q 0 ζ 0 /G x Q p 1 (x), if ζ 0,..., ζ i T x, then ζ 0,..., ζ i, ζ i+1 T x q i+1 ζ i+1 /G x Q q i ζ i /G x. The function x T x 2 (A<ω) is Borel. Since Q forces that each Ḋn is open dense, the tree T x has no terminals for each x B p 0. We consider the function f: B p 0 A ω such that for x B p 0, f(x) is the left-most branch of T x. Note that f is Borel. Indeed, since T x has no terminals, being its left-most branch is expressible with quantifiers over A <ω only. Let y x = {s qn x : n ω}, where q n x = q ζ n f(x)(n) /G x. Then y x is Mathias over N[G x ]. Finally, we define p 1(x) = s p1(x), y x \ s p1(x). 3.3 Iteration of length ω Now we will sketch an argument that P ω = Qω to give a brief glance at the limit step of iteration. If p P ω then p = p(i): i < ω is such that each p(i): ([ω] ω ) i Q is Borel. As before, we denote q(i) by q i, p(i) by p i, and we treat p 0 simply as a condition in Q rather than a function. Recall that we have defined the Borel set B p = { x ([ω] ω ) ω : ( i < ω) x i B pi }, where the Borel sets B p i ([ω] ω ) i, i < ω, are given by: B p 0 = { } x 0, x 1,..., x i B p i+1 iff x 0, x 1,..., x i 1 B p i and where q = p i ( x 0, x 1,..., x i 1 ). s q x i s q A q,

32 20 CHAPTER 3. ITERATION For p = p i : i < ω, p = p i: i < ω from P ω we have p Pω p B p B p ( x B p )( i < ω) p i ( x i) Q p i( x i). Recall that if q Q ω then q = q i : i < ω is such that q 0 Q and for each i < ω q 0, q 1,... q i Qi q i+1 Q. The function π: P ω Q ω is given as follows: π( p i : i < ω ) = q i : i < ω, where q 0 = p 0 and Q i+1 q i+1 = p i+1 ( ṙ j : j i ) for i < ω. To show that P ω = Qω we need to prove analogous facts to those from the previous section. Fact π: (P ω, Pω ) (Q ω, Qω ) is order preserving. Proof. One can show that for p P ω, Q ω π(p) Ġω ṙ i : i ω B p (c.f. Fact 3.11). Hence, if p Pω p, then π(p) Qω π(p ). Lemma The image π[p ω ] is dense in (Q ω, Qω ), i.e. for each q Q ω there exists p = p i : i < ω P ω such that π(p) Qω q. Proof. Take a countable N H κ with q = q i : i < ω N. Suppose that x = x i : i < ω ([ω] ω ) ω is Q ω -generic over N, i.e. G xi : i < ω is a Q ω -generic filter over N. Moreover, let q G xi : i < ω. For j < ω define p j ( x j) = q j / G xi : i < ω, i.e. the evaluation of q j in the model N[ G xi : i < ω ]. Note that if x = x i: i < ω ([ω] ω ) ω is Q ω -generic over N such that x j = x j then q j / G xi : i < ω = q j / G x i : i < ω, since q j is a Q j -name. If x 0, x 1,..., x i 1 can not be extended to an N-generic sequence with q in its induced filter, put p j ( x 0, x 1,..., x j 1 ) =, ω. Now we present the analogue of Lemma Lemma Suppose that p, p P ω and π(p) Qω π(p ). There exists p P ω such that p Pω p and p Pω p.

33 3.4. ITERATION OF ARBITRARY LENGTH 21 Proof. Let p, p N, where N H κ is countable. By Lemma 3.4 (with α = ω), there exists p P ω such that each x B p is Q ω -generic over N. Then, similarily as in the proof of Lemma 3.14, one can show that p Pω p and p Pω p. Arguing as in the Corollary 3.16 we get Corollary (P ω, Pω ) = (Q ω, Qω ). 3.4 Iteration of arbitrary length Now we will generalize the concept from Section 3.1 to arbitrarily long iterations of the Mathias forcing. Let us introduce the notation. If a a are countable and B ([ω] ω ) a, then B a is the projection of B onto ([ω] ω ) a, i.e. B a = { x a: x B}. If i: a a is a bijection and B ([ω] ω ) a, then B i = { x i: x B} ([ω] ω ) a. Fix an ordinal ε. Take a [ε] ω. We define a set P a as follows: p P a iff p is a function, dom(p) = a and for each ξ a the value p(ξ) is a code for a Borel function p(ξ): ([ω] ω ) a ξ Q. For p P a we define the Borel set B p ([ω] ω ) a of sequences which satisfy p. The definition proceeds by induction on ε (at the ε-th step of induction we deal with all possible P a, such that sup(a) = ε). Suppose that ε = max(a). In this case x ξ : ξ a B p x ξ : ξ a ε B p a ε & s x ε s A, where s, A = p(β)( x ξ : ξ a ε ). Suppose that ε = sup(a) is limit. Then x ξ : ξ a B p ( γ a) x ξ : ξ a γ B p a γ. Definition For ε ω 1 let P ε = {P a : a [ε] ω } with the ordering p Pε p dom(p) dom(p ) & B p dom(p) B p.

34 22 CHAPTER 3. ITERATION Note that if we use this definition for ε = α < ω 1 then the resulting poset is not equal to the one from Definition 3.1. However, the old one is a dense subset of the new one. We shall rather use the old one, i.e. for α < ω 1 we shall tacitly assume that p P α implies dom(p) = α. Note also that for a [ε] ω, P a itself can be considered as a forcing notion (with the ordering p Pa p iff B p B p ). Then P a is isomorphic to P α, where α = o.t.(a). Although P a P ε, it is not a complete subposet unless a = α. Example P {1} is not a complete subposet of P {0,1}. Let A 0, A 1,... be pairwise disjoint infinite subsets of ω. Let A be a maximal almost disjoint family of infinite subsets of ω with {A i : i ω} A. Define A = {A [ω] ω : ( A A ) A A < ω}. For A A let p A be such that p(1)( ) =, A. Then {p A : A A} is predense in P {1}. Let p P {0,1} be such that p(0)( ) =, ω and p(1)(x) =, A x, where A x is obtained as follows: By induction we define natural numbers n i for i ω. Let n 0 = min(a 0 /x(0)) (recall that x ω ω is the ennumeration of x) and n i+1 = min(a i+1 / max{x(i + 1), n i }). Define A x = {n i : i ω}. We claim that for every A A, p A is incompatible with p in P {0,1}. Indeed, suppose that we have p P {0,1} and A A such that B p B p and B p {1} B pa. We have two cases: either or ( i ω) A A i < ω ( i ω) A A i < ω. The first one is impossible: For every x [ω] ω we have A x A i = 1, hence A x A is finite. However, if x, y B p then y A x, so y / B pa. Therefore, we must be in the second case. Let f ω ω be given by f(i) = max(a A i ). Take x, y B p such that x dominates f. Again, we have that A x A is finite, so y / B pa. A contradiction. Let a [ε] ω, let α = o.t.(a) and let i a : α a be the increasing ennumeration of a. The cannonical isomorphism ι a : P α P a is given by ι a (p) = p, where p is a function on a, such that for each β < α, p(β) = p (i a β). Observe that for p P α B p = B ιa(p) i a = {x i a : x B ιa(p)}.

35 3.4. ITERATION OF ARBITRARY LENGTH 23 Fact If a a [ε] ω and p P a B p B p a. then there exists p P a such that Proof. Define p as follows: For ξ a let p(ξ)( x ζ : ζ a ξ ) = p (ξ)( x ζ: ζ a ξ ), where x ζ are defined by induction on ζ a in the following way: Suppose that we have x η for every η a ζ. If ζ a then x ζ = x ζ. If ζ / a then x ζ = sq A q where q = p (ζ)( x η: η a ζ ). It is easy to verify that B p B p a. Definition Define π: P ε Q ε, π(p) = q such that: if 0 / dom(p) then q(0) =, ω, if 0 dom(p) then q(0) = p(0)( ), if 0 < γ / dom(p) then q(γ) is a Q γ -name for, ω, if 0 < γ dom(p) then q(γ) is a Q γ -name such that Q γ q(γ) = p(γ)( ṙ ξ : ξ dom(p) γ ). A straightforward argument by induction on ε gives the next fact. The successor step is similar to Fact 3.11, the limit step is trivial. Fact Q ε π(p) Ġε ṙ ξ : ξ dom(p) B p. Arguing as in Corollary 3.12 we get Corollary π preserves the ordering, i.e. for every p, p P ε p Pε p π(p) Qε π(p ). The next lemma is a modification of [21, Lemma 3.8]. We present it without proof. One can get it following the argument from [21], which in fact resembles the proof of preservation of properness under countable support iteration. We just point out that one can repeat the argument from Lemma 3.17 to deal with a successor step.

36 24 CHAPTER 3. ITERATION Lemma 3.28 (c.f. [21, Lemma 3.8]). Suppose that N H κ is countable, ε N is an ordinal, and q Q ε N. Let a = ε N. Then there exists p q P a such that each x B pq is Q ε -generic over N with q in its induced filter, i.e. there is a (necessarily unique) G x Q ε N such that Q ε -generic over N, q G x, and x = ṙ ξ : ξ < ε /G x = ṙ ξ /G x : ξ a. In particular π(p q ) Qε q is N-master. Using this lemma we get: Lemma π: P ε Q ε is a dense embedding. Proof. To show that the image of π is dense fix q Q ε and take a countable N H κ with ε, q N. Denote a = ε N and let p q P a be given by Lemma Then π(p q ) Qε q. We need to show that π preserves incompatibility. To this end it is enough to prove that for any p, p P ε, if π(p) Qε π(p ) then there exists p Pε p, p. Fix p, p P ε such that π(p) Qε π(p ). Let N H κ be countable, ε, p, p, π(p), π(p ) N and let a = ε N. Define q = π(p) and take p = p q as in Lemma We will show that p q Pα p. Let x B pq. Take G x as in Lemma 3.28 (so x = ṙ ξ /G x : ξ a and q G x ). Since q Qε π(p ) we have π(p ) G x. By Fact 3.26, N[G x ] = x B p and by absoluteness, x B p. Hence, B pq B p. Similarily, one can show that p q Pα p. Corollary For any ordinal ε we have P ε = Qε. From now on, we will rather use P ε than Q ε and we will treat Ġε, ṙ ξ as P ε -names. Note that P ε forces that ξ, ṙ ξ is the only member in the intersection of all B p {ξ}, such that p is in the generic filter and ξ dom(p). By Fact 3.26 we get: Corollary For any ordinal ε we have ) P ε (p Ġε ṙ ξ : ξ dom(p) B p.

37 3.4. ITERATION OF ARBITRARY LENGTH 25 The following is an analogue of Lemma 3.6. Lemma Let a [ε] ω and B ([ω] ω ) a be Borel. Then for every p P ε there exists p Pε p such that a dom(p ) and either (B p a) B or (B p a) B =. Proof. Take p Pα p be such that a dom(p ) and p Pε ṙ ξ : ξ a B or p Pε ṙ ξ : ξ a B c. Without loss of generality assume the former. Denote q = π(p ). Take countable N H κ, such that ε, p, q, a, B N. Let a = ε N. So, a a. Using Lemma 3.28 find p q, such that dom(p q ) = a and B pq consists solely of (Q ε ) N -generics over N. Define p = p q. Arguing as in the proof of Lemma 3.29, we get that p Pα p and B p a B. The next lemma is needed to prove the consistency of axioms, which will be introduced in forthcoming chapters. Lemma Suppose that ε is an ordinal, a [ε] ω, α = o.t.(a), i a : α a is the increasing ennumeration of a, ι a : P α P a is the cannonical isomorphism, p P a, and B ([ω] ω ) a is a Borel set. Then (p Pε ṙ ξ : ξ a B) ( ι 1 a (p) Pα ṙ γ : γ < α B i a ). Proof. We will show the right-to-left implication only; the opposite one is similar. Suppose that ι 1 a (p) Pα ṙ γ : γ < α B i a. Fix p Pε p. By Lemma 3.32 there exists p Pε p, such that either B p a B or B p a B =. If B p a B, then by Corollary 3.31, p Pε ṙ ξ : ξ a B. We will show that the other case is impossible. Assume a contrario that (B p a) B =. By Fact 3.24 get p + P a, such that B p + B p a. We have B p + B = and B p + B p, so p + Pa p. Therefore, ι 1 a (p + ) Pα ι 1 a (p) and B ι 1 a (p + ) (B i a) =. By the (remark below) Lemma 3.5 we have A contradiction. ι 1 a (p + ) Pα ṙ γ : γ < α / B i a.