THE CONSISTENCY OF ARBITRARILY LARGE SPREAD BETWEEN THE BOUNDING AND THE SPLITTING NUMBERS
VERA V. FISCHER
A DISSERTATION SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
GRADUATE PROGRAM IN MATHEMATICS YORK UNIVERSITY TORONTO, ONTARIO
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In the following K and A are arbitrary regular uncountable cardinals. What was known?
THEOREM 1 (Balcar-Pelant-Simon, [2]). It is relatively consistent with ZFC that s = Wi < b = K.
THEOREM 2 (Shelah, [31]). It is relatively consistent with ZFC that s = K < b = A.
THEOREM 3 (Baumgartner and Dordal, [7]). Adding K Hechler reals to a model ofGCH gives a generic extension in which s = U\ < b — K.
THEOREM 4 (Shelah, [31]). There is a proper forcing notion of size continuum, which is almost "uj-bounding and adds a real not split by the ground model reals.
THEOREM 5 (Shelah, [31]). Assume CH. There is a proper forcing extension in which b = uj\ < s = u>2.
THEOREM 6 (Brendle, [11]). Assume GCH. Then there is a ccc generic extension in which b = Wi < s = K.
THEOREM 7 (M. Canjar, [14]). I/O = c, then there is an ultafilter U such that My does not add a dominating real.
iv ABSTRACT v
THEOREM 8 (Velickovic, [36]). Let K > Ha be a regular cardinal. Then there is a ccc generic extension satisfying MA + 2K° = K together with the following statement: For every family D of 2Ho dense subsets of the partial order X of all perfect trees, there is a ccc perfect suborder VofX such that D D V is dense in V, for all D ET>.
What is new?
u THEOREM 9. // cov{M) = K and Ti C u is an unbounded, <*- directed family of size K, VA < K(2X < K), then there is a a-centered suborder of Shelah 's proper poset from Teorem 4, which preserves 7i unbounded and adds a real not split by the ground model reals.
THEOREM 10. Assume GCH. Then there is a ccc generic extension in which b = K < s = K+ .
The above result is an improvement of S. Shelah's consistency of b = u>i < s = u>2. In chapter V, see Definition 5.4.2, we suggest a countably closed, K2-c.c. forcing notion P which adds a cr-centered forcing notion of C(w2)-names for pure conditions Q(C), such that Q(C) preserves all unbounded families unbounded and adds a real not split by Vc^ fl [u]w. An appropriate iteration of the forcing notion could provide the consistency of b = « < s = A. To my parents, Teodora and Velizar
vi Acknowledgement
Special thanks are due to my supervisor, Dr. Juris Steprans for his support throughout my doctoral studies, enthusiasm and encour agement regarding the work on my dissertation. I would like to thank also Dr. Paul Szeptycki, Dr. Mike Zabrocki and Dr. Ilijas Farah, for helpful comments and suggestions during earlier discussions of this work. Special thanks are due also to my external examiner Dr. Claude Laflamme, as well as to the other members of the examining committee Dr. Stephen Watson and Dr. Jimmy Huang.
Last, but not least, I would like to thank Arthur and my family for their love and support.
Vll Contents
Abstract iv
Dedication vi
Acknowledgement vii
Chapter 1. Introduction 1
1.1. The Bounding and the Splitting Numbers 2 1.2. Forcing 7 1.3. A proper forcing argument 17
Chapter 2. Centered Families of Pure Conditions 27
2.1. Logarithmic Measures 27 2.2. Centered Families of Pure Conditions 31 2.3. Partitioning of Pure Conditions 33 2.4. Good Names for Reals 36 2.5. Generic Extensions of Centered Families 38 2.6. Preprocessed Conditions 40 2.7. Generic Preprocessed Conditions 42
Chapter 3. b = K < s = K+ 46 3.1. Induced Logarithmic Measures 46 3.2. Good Extensions 49 3.3. Mimicking the Almost Bounding Property 51 CONTENTS ix
3.4. Adding an Ultrafilter 53 3.5. Some preservation theorems 56
3.6. b = K
Chapter 4. Symmetry 64 4.1. Q{C) which preserves unboundedness 64 4.2. Symmetric Names for Sets of Integers 69 4.3. Symmetric Names for Pure Conditions 73 4.4. An ultrafilter of Symmetric Names 75 4.5. Extending Different Evaluations 81
Chapter 5. Preserving small unbounded families 86 5.1. Preprocessed Names for Pure Conditions 86 5.2. Induced Logarithmic Measure 91 5.3. Good Names for Pure Conditions 93 5.4. Unboundedness 94
Chapter 6. A look ahead 101
6.1. General Definition of Symmetric Names 101
6.2. The N2-chain condition 103 6.3. Conclusion and open questions 106
Bibliography 110 CHAPTER 1
Introduction
Before proceeding with a brief account of the historical develop ment of mathematical ideas which lead to the establishment of the cardinal invariants of the continuum as a separate subject, we will in troduce some basic notions and give the contemporary definitions of the bounding and the splitting numbers since they present the main object of study of this work. Following standard notation we denote by uu the set of all functions from the natural numbers to the nat ural numbers and by [uf the set of all infinite subsets of ui. Let / and g be functions in "w. The function / is said to be dominated by the function g if there is a natural number n such that / <„ g, i.e.
(Vi > n)(f(i) < g(i)). Then <*= \Jn€u! b = min{|S| : B C wco and B is unbounded}. If A, B £ [w]w and both of the sets A D B and AnBc are infinite, then A is said to be split by the set B. A family S of infinite subsets of u is l 1.1. THE BOUNDING AND THE SPLITTING NUMBERS 2 said to be splitting if for every A G [u]w there is B G S which splits A. Then the splitting number is defined as the minimal size of a splitting family. That is s = min{|5| : S C [uf and S is splitting}. Recall also that if A, B are subsets of UJ, then the set A is said to be almost contained in the set B, denoted A C* B if A\S is finite. 1.1. The Bounding and the Splitting Numbers With the development of analysis in the nineteenth century, emerged a necessity of better understanding of the set of irrational numbers and the properties of the real line. In 1871 answering a question of Rie- mann, Georg Cantor obtained uniqueness of the trigonometric series representation of a function. That is he showed that if two trigonomet ric series converge to the same function, except on finitely many points, then they must be equal everywhere. A year later, he generalized his result to infinite sets of exceptional points. Recall that if S C M, then the derived set S' of S consists of all limit points of S. Cantor showed that if two trigonometric series converge to the same function except on a set S such that for some n € N the n-th derived set S^ is finite, then the series must be equal everywhere. Although results concerning infinite sets of exceptional points were already presented in the literature by that time, for example in 1829 Dirichlet suggested that a function whose point of discontinuity form a nowhere dense set is integrable, Cantor's generalized uniqueness theorem was one of the 1.1. THE BOUNDING AND THE SPLITTING NUMBERS 3 first results that took extensive use of the structure of an infinite set (see [25]). The result was followed in 1874 by Cantor's proof that the real numbers can not be placed in bijective correspondence with the natural numbers, the surprising fact in 1878 as Cantor himself admits, that n-dimensional Euclidean space is in bijective correspondence with the real line and the continuum hypothesis, that is the hypothesis that every infinite subset of K. is either in bijective correspondence with the natural numbers or the real line. By 1879 the study of combinatorial structure of infinite sets of reals has already emerged as an important direction in further studies of the properties of the continuum. For ex ample, Cantor defined and studied perfect sets of reals, i.e. sets which contain all of their accumulation points [25], and showed that every perfect set is in bijective correspondence with the real line. However later Bernstein constructed an uncountable set, such that neither it nor its complement contained a perfect set, thus it became clear that the study of perfect sets was insufficient to settle the continuum hypothesis. The cardinal invariants of the continuum arise from various combi natorial structures on the real line. Of particular interest for this work is the covering number of the meager ideal cov(M). Let M. denote the family of all meager subsets of R. Then cov(M) is the minimal size of a family T C M. such that (J T = M.. In 1899 Rene Baire showed that countably many meager sets do not cover the real line. Under the CH the minimal size of a family of meager sets which covers the real line is Hi = c. However if CH fails and for example c = N2, then it is consistent with ZFC that cov(.M) = Nj and also it is consistent that 1.1. THE BOUNDING AND THE SPLITTING NUMBERS 4 cov(A4) = K2. Another cardinal invariant which should be mentioned is the dominating number 5. A family D C wu is said to be dominating if for every function / € ww there is d 6 D such that / <* d. The dominating number X) is the minimal size of a dominating family. Note that every dominating family is unbounded and so b < D. In his "Calculus of Infinity" (see [18]) Paul du Bois-Reymond in the late 1870's studied the collection of continuous, monotone increasing positive valued functions and suggested to rank them according to their rate of divergence, or convergence to zero. That is he wanted to find a linear order -< on this set of functions, and an equivalence relation ~ such that / -< g provided that the "rate of growth of / is smaller than the rate of growth of g" and / ~ g provided they have the same rate of growth, and such that the equivalence ~ respects -< (see [29]). He defines / -< g if lim f(x)/g(x) = 0 or lim g(x)/f(x) = 00 x—>oo x—>oo and / ~ g if 0 < lim f{x)/gix) < 00. x—>oo The major problem in this ranking is the existence of incomparable infinities, that is the existence of functions for which the above limit does not exist. Cantor considered this a significant drawback of du Bois-Reymond's idea, a drawback which under the axiom of choice his cardinal arithmetic did not have. However, Hausdorff further pursued the study of maximal linearly ordered subsets of (NE, <*). In 1909 Hausdorff ([22]) showed that these maximal linearly ordered subsets 1.1. THE BOUNDING AND THE SPLITTING NUMBERS 5 have the cardinality of the continuum and established his main result the existence of (wi.o^-gaps. However it was not until 1936 (see [23]), that Hausdorff published the proof for binary sequences, i.e. estab W lished that in (2 , <*) there is a ( In fact, the first to give the contemporary definition of the bound ing number is Rothberger. A subset A of the n-dimensional Euclidean space Rn is said to have the property A, if each of its countable sub sets is relative G$- A subset A of K™ is said to have the property A' if A U B has the property A for every countable subset B of R™. Answering a question of Sierpinski, Rothberger constructs a set which has the property A, and at the same time does not have property A'. In [28] he defines -B(Kf) to be the proposition that all sequences of natural numbers of cardinality N^ are bounded, and then defines Hv to be the minimal cardinal for which B(^) does not hold. Thus in contemporary notation Hv is the cardinal invariant b and Rothberger's result states that a subset A of M" has the property A' if and only if the cardinality of A is less than the bounding number. By the time the splitting number appeared in the literature, the dependence of the topological, measure theoretic properties of the continuum and its car dinal combinatorial characteristics was well established. In fact the splitting number appeared as an algebraic characteristic of sequential 1.1. THE BOUNDING AND THE SPLITTING NUMBERS 6 compactness. In [10] David Booth states that for every regular un countable cardinal X, the space 2A is sequentially compact if and only if for every sequence {aa : a E A) of infinite subsets of N there is b CN such that for all a E X, b C* aa or b C* N — aa. In contemporary notation that is, 2A is sequentially compact if and only if A is smaller than the splitting number. Below is a list of the known consistency relations between the bounding and the splitting numbers, as well as the main results of this work; « and A denote arbitrary regular uncountable cardinals. THEOREM 1.1.1 (Balcar-Pelant-Simon, [2]). It is relatively consis tent with ZFC that s = u>\ < b = K. THEOREM 1.1.2 (Shelah, [31]). It is relatively consistent with ZFC that s = K < b — X. THEOREM 1.1.3 (Shelah, [31]). It is relatively consistent with ZFC that b = u)\ < s = u>2. THEOREM 1.1.4 (Brendle, [11]). It is relatively consistent with ZFC that b = u>i < s = K. Our main result, is the following. THEOREM 1.1.5 (Main result). It is relatively consistent with ZFC that b — K < 5 = K+ . In chapters IV-VI we give a first step towards the proof of the relative consistency ofb = K 1.2. Forcing In 1963 Paul Cohen introduced the method of forcing (see [15]) to obtain the independence of the continuum hypothesis. Since then the method of forcing is largely used to obtain different relative con sistency results, including results regarding the combinatorial cardinal characteristics of the real line. This is a general method for obtaining models of large finite fragments of ZFC, which satisfy some additional axioms. Excellent exposition of the method of forcing can be found in [24], [20]. I will give some basic notions and outline some of the fundamental properties of the method of forcing, since this is a major technique for the presented work. A forcing notion is a partially or dered set, that is a set P together with a reflexive and transitive relation < on P. We will work with strong forcing notions, i.e. partial orders which are also antisymmetric. That is for all p, q G P if p < q and q < p then p = q. The elements of the forcing notion are called conditions. If p < q then p is said to be an extension of q, also to be stronger than q and q is said to be weaker than p. The intuitive idea is that stronger conditions have more information about the intended model than weaker conditions. Conditions which do not have common ex tensions are said to be incompatible and respectively conditions which have a common extensions are said to be compatible. A set D C P is dense if every condition p E P has an extension in D. A set A C P is an antichain if its elements are pairwise incompatible. A set D C P is pre-dense if every element of the forcing notion is compatible with some d G D. A set C C P is centered if for all p,q € C there is r € C 1.2. FORCING 8 which is their common extension. A centered GCP which is closed with respect to weaker conditions is a filter. In the following c.t.m. ab breviates "countable transitive model of sufficiently large finite portion of ZFC". DEFINITION. Let V be a c.t.m., F e V a forcing notion. A filter GCPis generic over V, it GHD ^ $ for all dense DCF, D GV. Equivalently, in the above definition one can require that the filter G meets all pre-dense sets, or all maximal antichains, or all dense open sets which belong to the model V. Recall that a dense open set, is a dense subset which is closed with respect to stronger conditions. THEOREM. Let 7bea c.t.m., Pel^a forcing notion and GCP a filter generic over V. Then there is a countable transitive extension V[G] of V which contains G, has the same ordinals as V and is minimal, in the sense that if W is a transitive extension of V such that G € W, then V[G] C W. The model V is called the ground model and V[G] the generic ex tension. We will be working with forcing notions, which have the property that every element has incompatible extensions. Such posets are known to provide generic extensions which are distinct from the ground model. Indeed, it is not hard to verify that for such forcing notions the generic set does not belong to the ground model (see [24]). The elements of the generic extension have names in the ground model, which are recursively defined relations. 1.2. FORCING 9 DEFINITION. Let P be a forcing notion. Then X is a F-name if X is a relation and for all (Y,p) e X, Y is a P-name and p £ P. The collection of all P-names is a proper class. However if V is a c.t.m. and P € V, then the collection V¥ of all P-names in V is a set. The notion of a P-name is absolute and so V¥ = {X : (X is a P name)v}. The generic filter determines an evaluation of the names. More precisely if X is a P-name and G is a P-generic filter then the set X[G] = {Y[G} : 3p e G((Y,p) e X)} is the evaluation of X determined by G. Furthermore V[G] = {X[G\ : X € V¥}. With the forcing notion we associate a forcing language which is an exten sion of the language of set theory. An important characteristics of the forcing extension is the fact that there is a clear relationship between its semantic properties and the forcing notion, given by the forcing re lation. The forcing relation II- is a relation between the elements of the forcing notion and the sentences of the forcing language. This relation is definable in the ground model and gives a description of the generic extension within the ground model. The following statement is known as the forcing theorem. THEOREM. If V is a c.t.m., P e V is a forcing notion, 0 is a sentence in the forcing language and G C P is a filter generic over V then V[G)t= We would like to obtain generic extensions in which oj\ < b < s. Note that prior to this work, the existence of such models was un known. In 1984, see [31], S. Shelah obtains a generic extension in which b = toi < s = o>2- About 15 years later modifying a model of A. Blass and S. Shelah which gives an arbitrarily large spread between u and D, J. Brendle obtains the consistency of b = u)i < s = K for K arbitrary regular uncountable cardinal (see [31] and [11]). For our purposes, it is particularly important to obtain generic extensions in which cardinals are not collapsed. Observe that the notion of a cardinal is not absolute. For example forcing with the partial order of all finite partial functions from UJ to u\ with extension relation reverse inclusion, over a model M of CH, produces a generic extension in which u^ is a countable ordinal. In between the partial orders known to produce generic extensions in which cardinals are not collapsed, are the ccc forcing notions and the proper forcing notions. A forcing notion is said to be ccc, that is to satisfy the countable chain condition, if it does not contain uncountable antichains. A forcing notion is proper, if for every uncountable cardi nal A, every stationary subset of [A]" from the ground model remains stationary in the generic extension. The class of ccc forcing notions is contained in the class of proper forcing notions (see [30]). The method of forcing can be repeated in the generic extensions, which leads to the theory of iterated forcing, an excellent exposition of which can be found in [6]. There are certain preservation theorems concerning finite sup port iterations of ccc forcing notions, which present key points in the construction of models of io\ < b < s. For example, the finite support 1.2. FORCING 11 iteration of ccc forcing notion is ccc and so finite support iterations of ccc forcing notions do not collapse cardinals. In particular the finite support iteration of ccc forcing notions can be used to obtain generic extensions in which cardinals are not collapsed and the continuum is arbitrarily large. Another important fact is that if H is an unbounded family of functions in wu;, every countable subfamily of which is dom inated by an element of TC, then in order to preserve TC unbounded along an iteration with finite supports of ccc forcing notions, it is suffi cient to preserve TC unbounded at each successor stage of the iteration (see Theorem 3.5.2). Note that iterations of proper forcing notions of length > UJ2 are known to collapse the continuum. In general, there are few available iteration techniques leading to generic extension in which No H 2 > K3 (or even 2 ° > H2) and on the other hand there are many longstanding open questions about the combinatorial properties of the real line, whose solution would require models with continuum > H3. In between those are (see for example [33]) • the consistency of p < t • the consistency of no P-point and no Q-point • the consistency of s being singular Thus to a certain degree, results about the combinatorial charac teristics of R leading to generic extensions with large continuum, might be considered test results for developing new iteration techniques. The bounding and the splitting numbers are independent, that is it is consistent with ZFC that b < s as well as s < b. The consistency of s < b is first mentioned by Balcar, Pelant and Simon [2] in 1980. 1.2. FORCING 12 In 1984 S. Shelah obtains a different model of s < b (see [31]) and in 1985, [7] J. Baumgartner and P. Dordal show that in the Hechler model (i.e. a model obtained as a finite support iteration of Hechler forcing of length K, for K regular uncountable cardinal over a model of GCH) the bounding number is K while the splitting number remains oj\. In order to obtain a generic extension in which b < s one has to accomplish two major tasks: preserve a given unbounded family unbounded and increase the splitting number. By the preservation theorem mentioned above, in order to preserve an unbounded family unbounded along a finite support iteration of ccc forcing notions, it is sufficient to preserve the family unbounded at successor stages of the iteration. On the other hand in order to increase the splitting number along such an iteration, it is sufficient coflnally often to add reals which are not split by the ground model reals. A forcing notion which is known to add a real not split by the ground model reals is Mathias forcing [26]. DEFINITION. Mathias forcing M consists of all pairs (s, A) where s is a finite subset of u and A 6 [w]w such that max s < min A. We say that p1 = (si,Ai) < P2 = (s2, A2) if s\ end-extends s2, S\\s2 Q A2 and A\ C Ai. If S\ = S2 then p\ is said to be a pure extension oip2- If p = (s, A) is a Mathias condition, then the infinite set A is called the pure part of p and the finite set s the stem of p. Having in mind the notion of preprocessed conditions, observe that every extension can be obtained in two steps: extension of the stem followed by a pure extension. To see that M adds a real not split by the ground model reals, consider any A € [CJ]W PI M and p = (s, B) G M. Then B n A or 1.2. FORCING 13 B n Ac is infinite. That is for every A E [cu]" n M, the set c DA = {(s, B):B Df = {(s, A) : W G coA(£) > f(\s\ + £)} is dense in M. Indeed, given (s, B) E M, one can recursively define an infinite subset A of B so that (s, A) E Df. Then GC\Df contains some condition (s, A). Since UQ\S C A and s is an initial segment of UG, for every £ E oo fa(£+\s\)>A(e)>f(\s\+£). 1.2. FORCING 14 That is f <* fa- Therefore an iteration of M with countable supports of length u)2, as suggested above, would produce a generic extension in which the bounding number is also 102- By adding additional combinatorial structure on the pure Mathias conditions, in [31] S. Shelah obtains a forcing notion Q' (see defini tion 1.3.5) of size c which is proper, in fact Axiom A, which adds a real not split by the ground model reals and satisfies a strong combinatorial property. This property guarantees that under an iteration of the forc ing notion Q' with countable supports over a model of CH, the ground model reals will remain unbounded and so a witness to b = u)\. There fore an iteration with countable supports of length u2 over a model of CH of Shelah's forcing notion Q' produces a generic extension in which b = u>i < s = LU2- The additional combinatorial structure on the pure Mathias conditions, is given in the form of logarithmic measure on the finite subsets of u. The partial order Q' consists of pairs p = (u, T) where u is a finite subset of u and T = {(xj,/ij) : i € OJ) is an infinite sequence of finite logarithmic measures of strictly increasing levels. The sequence T is similarly to Mathias forcing called a pure condition, also pure part of p. Note that if int(T) = U{XJ : i G u>}, then (u, int(T)) is a Mathias condition. The properties of the finite logarithmic measure imply that 1.2. FORCING 15 if T = ((xi,hi) : i € w) is a pure condition and yl C a; is infinite, then either (hi(Xi D A) : i G to) ox {hi(xi D Ac) : i G u) is unbounded. Therefore T has a pure extension R such that int(i?) C A or int(i?) C c A and so for every A G [uf n M the set DA = {(u,T) : int(T) C A or int(T) C Ac} is dense. This implies that Q' adds a real not split by the ground model reals. However, we would like to obtain a model of u\ < b < s and so we would need to produce a generic extension in which cardinals N are not collapsed and c = 2 ° > N3. A partial order P which can be presented in the form P = [Jn€uJXn where for all n G UJ1 Xn is a centered subset of P is called a-centered. Note that every a-centered forcing notion has the countable chain condition. For every ultrafilter U let M[/ denote the suborder of Mathias forcing notion M consisting of all conditions whose pure part belongs to U. Conditions in My with equal stems are compatible and so My is cr-centered. Using the fact that U is an ultrafilter, one can easily show that My adds a real not split by the ground model reals. Therefore My can be used to obtain generic extensions in which the splitting number is arbitrarily large. However My might add a dominating real. In fact if U is selective, then forcing with My does add a real dominating the ground model reals. In [14] M. Canjar shows that if U is an ultrafilter such that My does not add a dominating real, then U is necessarily a F-point with no Q-points below it in the Rudin-Keisler order. In [14] it is also shown that if D = c then there is an ultrafilter U such that My does not add a dominating real. One may expect that an appropriate iteration 1.2. FORCING 16 of such My's would produce a generic extension in which b < s. For example given regular uncountable cardinals K < \ begin with a model of GCH, add n Hechler reals to obtain a generic extension M in which b = D — c = K and iterate with finite support of length A Canjar's My which do not add dominating reals. Note that along this iteration small dominating families are not introduced and in fact in each initial stage of the iteration d = c. Then in the final generic extension D = s = c = A. However preserving the ground model reals unbounded is not sufficient to preserve a given unbounded family unbounded along such an iteration and so one can not preserve a witness for b = K. In chapters II and III of this work we generalize Shelah's result to models of b = K < s = n+ for n arbitrary regular uncountable cardinal. In fact given an unbounded family 7i of functions in wui such that every subfamily H! of cardinality strictly smaller than \7i\ is dominated by an element of Ji (such families are called <*-directed) we will obtain a centered family C-n of pure conditions in Shelah's partial order Q' and a ccc suborder Q(Cn) of Q', which generalizes the relativization My of Mathias forcing to an ultrafilter U. The forcing Q{Cn) has the advantage to Canjar's non-dominating My that it not only adds a real which is not split by the ground model reals and preserves the ground model reals unbounded, but also preserves the given unbounded family Ti unbounded. Then under an appropriate finite support iteration of ccc forcing notions we obtain the consistency of b = « < s = K+. There are certain conditions on the existence of the forcing notion Q(Cn), one of which is that \H\ = 2H° and so this method can not be used to obtain 1.3. A PROPER FORCING ARGUMENT 17 generic extension in which o»i < b = K < K++ < s. In the second half of this work, we suggest a generic analogue of Mj/, in fact a generic analogue of <3(C), which has the countable chain condition, adds a real not split by the ground model reals and preserves a chosen unbounded family H of cardinality strictly smaller than 2No unbounded (in fact we will obtain slightly more). Thus the suggested forcing notion has the potential of providing a generic extension in which the splitting number s is arbitrarily larger than b > u>i. 1.3. A proper forcing argument Having in mind certain analogies between Shelah's model of b — + u>i < s = cu2 and the model of b = K < s = K (sections 2.1 - 3.6), in this section we give a more detailed outline of Shelah's proof. Apart from the original paper [31] (and [30]) an excellent presentation of this material is given in [1]. The section is self-contained and the rest of the work does not depend on it. Recall that a forcing notion P is weakly bounding if the ground model reals remain an unbounded family in every generic extension via P. However, the iteration of weakly bounding forcing notions is not necessarily weakly bounding (see [1]) and so a stronger notion of unboundedness is needed - see [31]: DEFINITION 1.3.1 (Shelah, [31]). The partial order P is almost "u- bounding if for every P-name / for a function in wu; and every condition peP there is a ground model function g 6 uui such that for every 1.3. A PROPER FORCING ARGUMENT 18 infinite subset A of u, there is an extension q& of p such that qA lh 3°°k E A(f(k) < g(k)). As mentioned in [31], the Cohen forcing notion is almost "^-bounding. By [30] the countable support iteration of proper almost wtu-bounding forcing notions is weakly bounding, and so in such iterations the ground model reals remain an unbounded family. It remains to observe the fol lowing preservation theorems (see [30] or [1]): THEOREM 1.3.2 (Shelah, [30]). Assume CH. Let {(P* :i<6), (Q* : i < 6}) where 5 < u2, he a countable support iteration of proper forcing notions of size Ni. Then CH holds in V¥s. THEOREM 1.3.3 (Shelah, [30]). Assume CH. Let (P* : i < S) where S < LO2, be a countable support iteration of proper forcing notions of size Ki. Then ¥$ satisfies the H2-chain conditions. Therefore beginning with a model of CH and iterating with count able support (of length u2) proper forcing notions of size continuum, which satisfy the almost ^cu-bounding property, one can obtain a generic extension in which cardinals are not collapsed and the ground model reals remain an unbounded family. Furthermore if the forcing notion adds a real which is not split by the ground model reals, such an it eration would give the consistency of b = oj\ < s = U2- Thus it is sufficient to obtain the following theorem. THEOREM 1.3.4 (Shelah, [31]). Assume CH. There is a proper, almost "co-bounding forcing notion Q' of size Kj such that in every 1.3. A PROPER FORCING ARGUMENT 19 (V,Q')-generic extension there is an infinite subset of u which is not split by the ground model reals. For completeness we give the definition of Shelah's partial order Q!. The partial order (defined in section 2.2) differs slightly from the forcing notion given below. DEFINITION 1.3.5 (Shelah). Let Q' be the set of all pairs (u,T) where u is a finite subset of to and T = ((si; /ij) : i G to) is a sequence of finite logarithmic measures such that (1) maxw < minso (2) max Si < min si+i (3) h^si) < hi+1(si+1). Also int(T) = U{SJ : i G to} denotes the underlying subset of u>. We say that (ui,Ti) is extended by (u2,T2) where Ti = (tf : i G to) for £ = 1,2, t\ = (4,hf) and denote this by (u2,T2) < (ui.Ti) if the following conditions hold: (1) U2 is an end-extension of ui and U2\ui C int(Ti) (2) int(T2) C int(Ti) and there is a sequence (Bi : i G to) of finite subsets of to such that maxti2 < minsj for j = min J90, max5j < minBj+i and s^ C U{s] : j G £»} (3) for every e C s? such that /if (e) > 0 there is j G 5, such that h)(er\s)) >0. Whenever (u, T) G Q' the finite set u is called the stem of the condition and T the pure part. Conditions with empty stem are called pure 1.3. A PROPER FORCING ARGUMENT 20 conditions and are often denoted by their pure part. If q extends p, and q has the same stem as p, then q is called a pure extension of p. Observe that if (u,T) is a condition in Q', then (u, int(T)) is a condition in the Mathias forcing notion. In fact the reason that Q' adds a real not split by the ground model reals is the same as for Mathias forcing. To see that Q' adds a real not split by the ground model reals, note that if T £ Q' is a pure condition and A is an infinite subset of u, then there is a condition T' £ Q' extending T such that int(T') C A or int(T') C Ac. But then for every ground model infinite subset A of u> the set c DA = {(u, T)eQ' : int(T) C A or int(T) C A } is dense in Q' and so the real UG = (J{u:3T(u,T)eG} where G is Q'-generic filter is not split by the ground model reals. Note that if {Fa : a < 5} is a countable support iteration of proper forcing notions, where 5 is of uncountable cofinality, then any new real is obtained at some initial stage 8Q < 5 of the iteration. Furthermore if (Fa : a < u>2) is a countable support iteration of proper forcing notions, then any set of reals of cardinality uj\ is added at some (proper) initial stage of the iteration. Therefore, assuming Q' is proper, an iteration of Q' over a model of CH of length u>2, would result in a model of s = u>2- 1.3. A PROPER FORCING ARGUMENT 21 DEFINITION 1.3.6 (Baumgartner, [5]). A forcing notion P = (P, <) is said to satisfy Axiom A, if the following holds: (1) There is a sequence of partial orders { m < n and p, q are conditions in P such that p P (2)If{pn} nsur is a sequence of conditions in P such that pn+i ^n+i pn for every n, then there is a condition p such that p s for every n. The sequence {pn}neu> i called a fusion sequence and p is called £/ie fusion of the sequence. (3) For every D C P which is dense, and every condition p, for every n 6 ui there is a condition p' such that p' <„ p and a countable subset Do of D which is predense below p'. The forcing notion Q' satisfies Axiom A and so is proper (see [5]). To see that indeed Q' satisfies Axiom A, define a sequence of suborders { («2,T2) if Ui = u2 and (u2,T2) if Ui = u2 and for every j G i t] = i|. Then { In order to establish part (3) of Axiom A as well as the almost ^-bounding property, one needs the notion of preprocessed conditions (see [6], Section 8). Note that in Section 2.6 we work with a slight modification of this notion. DEFINITION 1.3.7. Suppose D C Q' is a dense open set. We say that p = (u, T) where T — (U : i G LJ) is preprocessed for D and k 6 u if for every subset v of k which end-extends u, (v, (tj : j > k)) has a pure extension in D if and only if (v, (tj : j > k)) belongs to D. The following three Lemmas show that, whenever D is a dense open set and p e Q' there is a pure extension q of p such that for every i£w, q is preprocessed for D and i. LEMMA 1.3.8. Let D be a dense open subset of Q' and k € u. If (u, T) is preprocessed for D and k, then any extension of (u, T) is also preprocessed for D and k. PROOF. Consider any extension (w,R) of (u,T) where R = (r, : iGw). Let v be a subset of k, which end-extends w and such that (v, (rj : j > k)) has a pure extension in D. Since R extends T, by definition of the extension relation on Q (rj : j > k)) is an extension of (tj : j > k). Therefore (v,(tj : j > k)) has a pure extension in D and since (u, T) is preprocessed for D and k, (v, (tj : j > k)) belongs to D (note that v also end-extends u). But D is open and 1.3. A PROPER FORCING ARGUMENT 23 since (v, (rj : j > k)) extends (v, (tj : j > k)), (v, (rj : j > k)) also belongs to D. O LEMMA 1.3.9. Let (u,T) e Q', k e u. Then {u,T) has a PROOF. Let T = {U : i G UJ). Fix an enumeration vi,...,Vj of all subsets of k which end-extend u. Consider (vi, {U : i > k)). If (vi, (U : i > k)) has a pure extension in D, denote it (v\, (tj : i > k)). If there is no such pure extension, let t\ — U for every i > k. In the next step consider (u2, (t] : i > k)). If it has a pure extension in D, denote it (t>2, (t? : i > k). If there is no such pure extension, then for every i > k let t? = t\. At the j'-th step we will obtain condition (VJ, (fj : i > k)). Then (u, (i? : i € to)) where for every i < k, tf = ij is a To see this, suppose (v, (tl : i > k)) has a pure extension in D where v C k, v end-extends u. Then v = vm for some m, 1 < m < j. Then at _1 step m, we must have had that (vm, (i™ • i > k) has a pure extension in D, and so we have fixed such a pure extension (vm, (t™ : i> k)) £ D. However since m — 1 < j, we have (% '• i > k) < (i™ : i > k). But D is open and so (u, (^ '• i > k)) is an element of £) itself. D LEMMA 1.3.10. Let D be a dense open set. Then any condition has a pure extension which is preprocessed for D and every i£u. PROOF. Let p = (u,T) be an arbitrary condition. By Lemma 1.3.9 there is a fusion sequence {pi}i£W such that p0 = p, pi+\ Then for every i € u we have that q Observe that g is obtained as the fusion of a sequence. This fact will appear very important in obtaining the almost-^o; bounding property, and in particular Lemma 1.3.11. With this we are ready to show that the forcing notion Q satisfies Axiom A, part (3). Let D be a dense open set and p an arbitrary condition. By Lemma 1.3.10 there is a pure extension q = (u, T) for T = (tj : j £ ui) which is preprocessed for D and every i£w. Since q is obtained as a fusion of a fusion sequence below p, for every n £ u, q D0 = {(v, (tj : j > i)) € D : v C i, i £ UJ, v end-extends u} is a countable subset of D which is pre-dense below q. To see this consider an arbitrary extension (v, R) of q. Since D is dense, (v, R) has an extension (v U w, R') in D. Note that (v U w, R') is a pure extension of (v U w, (tj : j > k)) for some k € ui such that w C k. However q is preprocessed for D and k, and so (v U w, (tj : j > k)) G D. Thus in particular (v U w, (tj : j > k)) belongs to Do and is compatible with (v,R). This establishes axiom A and so properness. The main technical tool in obtaining the almost "^-bounding property is the following Lemma - compare with section 3.3. u LEMMA 1.3.11. Let f be a Q-name for a function in ui and letp = (u,T) be an arbitrary condition in Q'. Then there is a pure extension (u, R) ofp where R = (ri : i € w), r{ — (XJ, ft) such that \/i G to, \/v C i 1.3. A PROPER FORCING ARGUMENT 25 which end-extend u and Vs C Xi such that g^s) > 0 there is wv C s such that (v U wv, (rj : j > i + 1)) lh f(i) = k for some k G u>. In order to obtain the pure condition R of the above Lemma, one has to consider logarithmic measures induced by positive sets (see De finition 2.1.4) and in particular to show that the logarithmic measure induced by the family Vk(T, D) where T = (ti : £ 6 u) is a pure con dition preprocessed for a given dense open set D and fcGw consisting of all finite subsets x of int(T) such that for some £ G u, x n int(fy) is positive and Vv C k3w C x such that (vUw,T) 6 .D, takes arbitrarily high values - compare with section 3.1. Because of the analogy with Theorem 3.3.2 we give a proof of the almost "w-bounding property of Q' - see [1]. THEOREM 1.3.12. The forcing notion Q' is almost UUJ-bounding. PROOF. Let / be arbitrary Q-name of a function and p a condition in Q. Let q = (u,T), where T = (ij : i G u) and U = (XJ, gi), be a pure extension of p which satisfies Lemma 1.3.11. Then for every i € w let g(i) be the maximal k such that there are v C i and w C int(tj) such that (v U w, (tj :j>i + l))tir f(i) = k. Consider any A G [w]w and let qA = (u, (U:iE A)). We claim that qA lh 3°°A; G A(f(k) < g(k)). Let n G u) and let (v, R) be an arbitrary extension of qA. Then by definition of the extension relation there is i G A such that io > n, v C i and s = int(R) fl int(tj) is such that gi(s) > 0. But then by Lemma 1.3.11 there is w C s such that (v U w, (tj • j > io + 1)) II- /(i) = A; and so (v U io, (tj : j > i + 1)) lh /(i) < g(i). However 1.3. A PROPER FORCING ARGUMENT 26 (v U w, R) extends (v U w, (tj : j > i + 1)) and so (v U w, R) lh f(i) < g(i). Note also that (v\Jw,R) extends (v,R). Then, since (v,R) was an arbitrary extension of qA, the set of conditions which force "3k G A(k > n A /(&) < g(k))v is dense below 94. Since n was arbitrary as well, we obtain qA lh 3°°k £ A(f{k) < g(k)). • CHAPTER 2 Centered Families of Pure Conditions 2.1. Logarithmic Measures The notion of logarithmic measure is due to S. Shelah. In our presentation of logarithmic measures and their basic properties (Defin itions 2.1.1, 2.1.4 and Lemmas 2.1.3, 2.1.7, 2.1.9, 2.1.10) we follow [1]. A0, At such that A = AQ U Au h{Ai) > h(A) - 1 for i = 0 or i = 1 unless h(A) — 0. Whenever s is a finite set and h a logarithmic measure on s, the pair x = (s, h) is called a finite logarithmic measure. The value h(s) — \\x\\ is called the level of x. DEFINITION 2.1.2. Whenever h is a finite logarithmic measure on x and eCiis such that h(e) > 0, we will say that e is h-positive. LEMMA 2.1.3 (Shelah). If h is a logarithmic measure and h{AQ U • • • U An^x) > £ + 1 then h(Aj) >£~j for some j, 0 < j < n - 1. DEFINITION 2.1.4 (Shelah). Let P C [LO] 27 2.1. LOGARITHMIC MEASURES 28 (1) h(e) > 0 for every e € [w] (2) /i(e) > 0 iff e € P (3) for ^ > 1, /i(e) > £+1 iff e e P, \e\ > 1 and whenever eo, ei C e are such that e = eo U ei, then /i(eo) > £ or /i(ei) > £. Then /i(e) = £ iff £ is the maximal natural number for which h(e) > £. The elements of P are called positive sets and /i is said to be induced by the positive sets P. DEFINITION 2.1.5. Let h be an induced logarithmic measure. Then h is said to be atomic, if there is a singleton {n} such that h({n}) > 0. REMARK 2.1.6. Prom now on we assume that all logarithmic mea sures are non-atomic. LEMMA 2.1.7 (Shelah). If h is a logarithmic measure induced by positive sets and h(e) > £, then for every a such that e C a, h(a) > £. EXAMPLE 2.1.8 (Shelah, [1] or [32]). Let P be the family of all sets containing at least two points and h the logarithmic measure induced by P on [u] An easy application of Konig's Lemma gives the following: LEMMA 2.1.9 (Abraham, [1]). Let P be an upwards closed family of finite non-empty subsets of u and h the induced logarithmic measure. Let £ > 1. Then for every subset A of u, if A does not contain a set 2.1. LOGARITHMIC MEASURES 29 of measure > £ + 1, then there are A0,Ai such that A — Ao U A\ and neither of Ao, A\ contains a set of measure greater or equal £. PROOF. If A is a finite set, then the given statement is the contra- positive of part 3 of Definition 2.1.4. Thus assume A is infinite. For every natural number k, let Ak = A n k and let T be the family of all functions / : m —> Uo -1 1 1 /i(a™) £ £ Let a™ = Am_! D ag and a™" = An-i D of. Then by Lemma 2.1.7 the measure of each of a™-1, a™-1 is not greater or equal -1 to £ and Am-i = ^4n(m—1) = a^^Ua™ . Therefore in m steps we can define finite sequences (ag : 0 < A; < m), (af : 0 < /c < m) such that for every fc, Afc = ag U aj, /i(ag) ^ £, /i(aj) ^ £ and VA; : 0 < k < m - 1 +1 k+1 ag C ag , a* C a . Therefore / : m - Uo Therefore by Konig's Lemma there is an infinite branch through T. k k Let / : u - UfcewP(>U) x V(Ak) where /(A:) = (ag, a*), a Ua = Ak, a = a etc., be such an infinite branch. Then if B0 — Ufcew oi -^I Ufceu> i 2.1. LOGARITHMIC MEASURES 30 we have that A = BQ U BI and none of the sets BQ, B\ contains a set of measure greater or equal i. Consider an arbitrary finite subset x of Bo- Then x C a* for some k G u>. But /I(OQ) ^ •£ and so h(x) ^ £ The same argument applies to B\. D LEMMA 2.1.10 (Abraham, [1]). (Sufficient Condition for High Val ues) Let P be an upwards closed family of finite subsets of u and h the logarithmic measure induced by P. Then if for every n £ u> and every partition of u into n sets to — AQ U • • • U An^\ there is some j < n — 1 such that Aj contains a positive set x (such that \x\ > 2), then for every natural number k, for every n £ ui and partition of u into n sets oj = AQ U • • • U An^i there is some j < n — 1 such that Aj contains a set of measure greater or equal k. REMARK. Note that if the measure of x is > 2, then \x\ > 2. However for non-atomic measures, ||x|| > 0 implies that \x\ > 2. PROOF. The proof proceeds by induction on A;. If k — 1 this is just the assumption of the Lemma. So suppose we have proved the claim for k = £ and furthermore that it is false for k = £ + 1. Then there is suc some n G uj and partition of ui into n sets u = A0 U • • • U Ai-i h that none of A0,..., An^i contains a set of measure greater or equal £ + 1. By Lemma 2.1.9 for each j € n — s there are sets A®, Aj none of which contains a set of measure greater or equal £ and such that Aj = A^uAj. Then u = A® U A\ U • • • U A^l_1 U A\_^ is a finite partition of u>, none of the elements of which contains a set of measure > £. This contradicts the inductive hypothesis for k — £. • 2.2. CENTERED FAMILIES OF PURE CONDITIONS 31 2.2. Centered Families of Pure Conditions DEFINITION 2.2.1 (Shelah). Let Q be the set of all pairs (u,T) where u is a finite subset of LU and T = ((s*, hi) : i E to) is a sequence of finite logarithmic measures such that (1) maxu < minso (2) max Si < min Sj+i for alii E LU (3) (hi(si) :i Ecu) is unbounded. The finite part u is called the stem of the condition p = (u, T), and T the pure part of p. Also int(T) = U{SJ : s € a;}. In case that u = 0 we say that (0,T) is a pure condition and usually denote it simply by T. We say that («i, Ti) is extended by (u2, T2), where 7^ = (tf : i E tu) l and tf = (s£, h s) for £ = 1,2, and denote it by (u2, T2) < (uu 71) if the following conditions hold: (1) it2 is an end-extension of tiX and U2\ui C int(Ti) (2) int(T2) C int(Ti) and furthermore there is an infinite sequence (5j : i E u) of finite subsets of a; such that max u2 < min s] for j = minJ50, max(-B,) < min(J3i+i) and sf C U{s] : j £ Bi}. (3) for every subset e of sf such that h?(e) > 0 there is j E Bi such that h)(e n s}) > 0. If Ui = if2, then (^2,7^) is called a pure extension of (ui, 7~i). The partial order Q' in Definition 1.3.5 differs with Q, in the re quirement that the sequence (/ij(sj) : i E tu) is strictly increasing rather then simply unbounded. However from every unbounded sequence one 2.2. CENTERED FAMILIES OF PURE CONDITIONS 32 can choose a strictly increasing subsequence and so the partial order Q' (see Definition 1.3.5) is dense in Q. DEFINITION 2.2.2. Let T = (U : i G u) be a pure condition. Then for every k G u, let ir(k) = min{i : k < minint(ij)} and let T\k = TjT(fc) = {U : i > ix(k)). Whenever u is a finite subset of 10 let T\u = TiT(maxu) and (u,T) = (u,T\u). DEFINITION 2.2.3. Let T be a family of pure conditions. Then Q{F) is the suborder of Q of all (u, T) e Q such that 3R e F(R < T). DEFINITION 2.2.4. A family of pure conditions C is centered if whenever X, Y G C there is R € C which is their common extension. We will be interested in centered families C of pure conditions and the associated partial order Q(C). LEMMA 2.2.5. Let C be a centered family of pure conditions. Then Q(C) is a-centered. PROOF. Any two conditions in Q(C) with equal stems have a com mon extension in Q(C). • From now on by centered family we mean a centered family of pure conditions, unless otherwise specified. Furthermore we assume that all centered families are closed with respect to final sequences, that is if C is a centered family and T 6 C then T\v G C for every v £ [u] LEMMA 2.2.6. Any two conditions of Q(C) are compatible as con ditions in Q(C) if and only if they are compatible in Q. 2.3. PARTITIONING OF PURE CONDITIONS 33 PROOF. Let p = (u, T) and q = (v,R). Suppose that p,q are compatible as conditions in Q. Let (w, Z) be their common extension in Q. Then in particular w is a common end-extension of u and v, which implies that u is an end-extension of v or v is an end-extension of u. Say u is an end-extension of v. Then u\v C w\v C int(i?). Since p and q belong to Q(C), by definition there are pure conditions T', R' in C such that V < T and R' < R. However C is centered, and so there is a pure condition Z' e C which is a common extension of T" and R' and so a common extension of T and R. But then (ix, Z') is a common extension of p and g from Q(C). • A pure condition, which is compatible with every element of a cen tered family, is said to be compatible with the centered family. If C is a centered family which contains a centered family C in its downwards closure, i.e. C C Q(C), then C is said to extend C. In particular if C C Q(C) and there is R G Q(C) such that VX € C'(Z < i?) we say that C extends C below R. 2.3. Partitioning of Pure Conditions LEMMA 2.3.1. Let (x,h) be a finite logarithmic measure andh(x) < n. Then x = U{xi : % €E 2™} where for every i € 2n, h(xi) = 0. PROOF. We give a proof by induction on n. Let n = 1. Then by definition of logarithmic measure there are sets Xo, X\ such that x = XQ U X\ and h(x0) ~£_ 1 and h(xi) ^ 1, that is x0 and x\ are not positive. Suppose we have proved the claim for every measure of level < n, where n > 2. Let (x, h) be a logarithmic measure of level < n +1. 2.3. PARTITIONING OF PURE CONDITIONS 34 Then, there are sets XQ, x-± such that x = x0 U x^ and h(xo) < n, h(xj) < n. By inductive hypothesis for £ G 2 xe = \J{x\ : i E 2"} where for all i £ 2™ h(x\) = 0. But then it is clear, that x can be presented as the union of 2 x 2" sets, each of which is of measure 0. • LEMMA 2.3.2. Let T = ((SJ, hi) : i E to) be a pure condition and let A be an infinite subset of u. If the sequence (hi(si D A) : i E u) is bounded, then T has no pure extension R with int(R) C A. PROOF. Suppose to the contrary that R is a pure condition in Q extending T such that int(i?) C A. Then there is (Bi : i G to) C [u] 1 M /^•(sj ) = 0. Then for every m G 2 let am = x n (U{s™ : j G £/}). M M Then a; = U{am : m G 2 } and so by Lemma 2.1.3 there is m E 2 M such that g(am) > (2 + 1) - m > 1. But then 3j G Be such that /ij(am fl Sj) > 0. However s™ = am fl Sj and so hj(s™) > 0 which is a contradiction. • REMARK 2.3.3. It is essential to work with non-atomic measures. If P = [o;] (({2n}, h \ {2n}) : n E to) is a sequence of finite logarithmic measures of measure 1, however amalgamating successive measures of T one can obtain a pure condition, and so in particular an unbounded sequence of finite logarithmic measures. DEFINITION 2.3.4. Whenever T = ((sj,/ij) : i E u) be a pure condition and A C u, let T \ A = ((si C\A,hi \ V{si (1 A)) : i E u). LEMMA 2.3.5. Let T = (ti : i E U>), where U = (sj,/ij), be a pure condition and ui = Ao U • • • U ^4„_i a finite partition of u. Then there is j E n such that T \ Aj is a pure condition. PROOF. Suppose not. That is, for every j E n there is Mj E LO such that hi(si (1 Aj) < Mj for every i E u. Let M = maxj6n Mj and let U = (si,hi) be a measure from T with hi(si) > M + (n + 1). Let si = Si PI ylj for every j E n. Then s* = s° U s£ U • • • U s™_1 is a partition of Si into n sets and so there is j E n such that hi(s\) > hi(si) — j = M+(n + l)-j>M + l>Mj which is a contradiction. • LEMMA 2.3.6. Let R be a pure extension of T and let A be an infinite subset of u>, such that R \ A and T \ A are pure conditions. Then R \ A is a pure extension ofT \ A. PROOF. Let R = {{xugi) : i e u), T = ((sj,/ij) : i E u). Since R is a pure extension of T, there is a sequence (Bi : i E u) C [co} 0, by definition of the extension relation there is j E Bi such 2.4. GOOD NAMES FOR REALS 36 that gj(eC\Sj) > 0. It remains to observe that ef)Sj = efl Sj DA. Thus R \ A is a pure extension of T \ A. • LEMMA 2.3.7. Let C be a centered family, T a pure condition com patible with C and u = AQ U • • • U An_i a finite partition of u. Then there is j En such that T \ Aj is a pure condition compatible with C. PROOF. Suppose the claim is not true and let / C n be the set of all indexes j G n for which T \ Aj is a pure condition in Q. By Lemma 2.3.5, 7^0. By hypothesis, Vj € / there is 7} G C such that Tj is incompatible with T \ Aj. However I is finite, C is centered and so there is X € C which is a common extension of (Tj : j E I). By assumption X and T have a common extension R E Q. Again by Lemma 2.3.5 there is an i E n such that R \ Ai is a pure condition. Furthermore by Lemma 2.3.6 R \ Ai 2.4. Good Names for Reals REMARK 2.4.1. We will use the fact that whenever / E VF D uui for some forcing notion P, then / has a P-name of the form / = u {((^ fP)>P) -PEAi,iE u;,fp E UJ} where for every i E to, Ai = A{f) is a maximal antichain. DEFINITION 2.4.2. Let C be a centered family of pure conditions and let / be a <2(C)-name for a real. Then / is a good name if for every centered family C extending C, / is a Q(C")-name for a real. 2.4. GOOD NAMES FOR REALS 37 REMARK 2.4.3. That is / is a good C/(C)-name for a real if and only if for every centered family C extending C and for every i € u, Ai(f) remains a maximal antichain in Q(C). LEMMA 2.4.4. Let C be a centered family of pure conditions and let f be a Q(C)-name for a real. Then the following are equivalent: (1) f is a good Q{C)-name for a real. (2) For every centered family C extending C such that \C'\ = \C\, f is a Q(C')-name for a real. PROOF. The implication from (1) to (2) is straightforward. To obtain that (2) implies (1) consider any centered family C" extending C such that \C'\ > \C\ and suppose that / is not a C;(C")-name. Then there is condition p = (u, T) € Q(C) which is incompatible with all elements of A = A(f) for some i € u. Note that C U {T} C Q(C). Inductively we will construct a centered family C" contained in Q(C) such that C U {T} C C" and \C"\ = \C\. Let C0 = C U {T}. Since Co C Q(C) for all X, Y € C0 there is Zxy e Q(C) such that Zxy < X and ZX,Y < Y. Let C0 = {ZX,Y • X, Y e C0} and let Cx = C0 U C'0. Suppose we have defined Cn = Cn_i U C'n_x C Q(C) where n > 1, such that for every X, Y G Cn-\ there is Z 6 Cn such that Z < X, Z < Y and |C„_i| = \Cn\. Then since Cn C Q(C) for all X,Y € Cn there is ZX,Y G Q(C") such that Zx,y < X and ZX)Y < Y. Then let C'n = {ZXtY : X, Y € C„} and let Cn+1 = Cn U Q. With this the inductive construction is complete. Then C" = Un€hJCn is a centered family of pure conditions containing Cu{T} and such that \C"\ = \C\, 2.5. GENERIC EXTENSIONS OF CENTERED FAMILIES 38 C" C Q(C'). Note that C is infinite, since by assumption C is closed with respect to final subsequences. By the hypothesis of (2), / is a Q(C")-name for a real and so Ai(f) is a maximal antichain in Q(C"). Since CU {T} C C", p E Q{C") and so there is a condition q £ Q{C") which is a common extension of p and an element q of Ai. But Q(C") C (5(C) and so g G <2(C"), which is a contradiction to p being incompatible with all elements of Ai. • COROLLARY 2.4.5. Let C be a centered family of pure conditions and let f be a Q(C)-name for a real. If there is a centered family C extending C such that f is not a Q(C')-name for a real, then there is a centered family C" extending C which has the same cardinality as C and such that f is not a Q(C')-name for a real. 2.5. Generic Extensions of Centered Families DEFINITION 2.5.1. Let Qfin denote the partial order of all finite sequences of strictly increasing finite logarithmic measures with the end-extension relation. That is, Q/m is the set of all sequences f = (r0,..., r„), n € u such that for every i < n, rj = (SJ, hi) is a finite logarithmic measure and for every i < n — 1 max(sj) < min(si+1) and hi(si) < hi+1(si+1). The level of the sequence f = (r0,..., rn) is the level of the highest measure r„, denoted also ||f||. Whenever fi and ?i are sequences in Qfin define fi < f2 if f2 is an initial segment of f\. 2.5. GENERIC EXTENSIONS OF CENTERED FAMILIES 39 DEFINITION 2.5.2. Let f = (r0,...,r„_i) be a sequence in Qfin where for every i £ n r, = (si; /ij). Then f extends the pure condition T = (ti : i £ u), ti = (Xi,gi) denoted f < T, if (1) int(f) = U{SJ : i £ n} C int(T) and there is a sequence (Bo,..., Sn-i) of finite subsets of CJ such that max Bi < min5i+i for every i £ n — 1, and S; C U{:rj : jf G A} for alli£n (2) for every i £ n and eCs; such that /i,(e) > 0 there is j £ Bi such that gj{e fl Xj) > 0. The finite logarithmic measure r = (s,h) extends the pure condition T = (ti : i £ u), denoted by r < T, if the sequence f = (r) extends the pure condition T. DEFINITION 2.5.3. Let T be a pure condition. Then P(T) is the suborder of Qfin consisting of all finite sequences f extending T. LEMMA 2.5.4. Let T be a pure condition. Then (1) \fk£uj the set Ek = {f £ P(T) : \f\ > k} is dense in P(T) . (2) For every pure condition X compatible with T and every n £ u, the set DT(X,n) = {f £ P(T) : 3^- £ f{rj < X and \\rj\\ > n)} is dense in P(T). PROOF. Let f e P(T). Since T\int(f) and X are compatible, there is a finite logarithmic measure z of level higher than ||f|| and n, which is their common extension. Then f~(z) extends f and is in DT(X, n). • COROLLARY 2.5.5. LetC be a centered family of pure conditions, T a pure condition compatible with C and G a P(T) -generic filter. Then 2.6. PREPROCESSED CONDITIONS 40 in V[G] there is a centered family C extending C below RQ = UG = (rj : i E u) (and so below T) which is of the same cardinality as C. PROOF. By Lemma 2.5.4.1 RQ is a pure condition of strictly in creasing finite logarithmic measures. For every X E C, n E ui the set DT(X,n) is dense in P(T) and so G n DT(X,n) ^ 0. Then Ix = (i '• fi < X) is infinite and so RQ A X = {rt : i € Ix) is pure condition which is a common extension of RQ and X. Furthermore if Y < X then IY Q Ix which implies RG A Y < RG A X. Therefore the family {RQ A X}XeC is centered. • 2.6. Preprocessed Conditions DEFINITION 2.6.1. Let C be a centered family of pure conditions, / a good Q(C)-name for a real, k, i £ cu and T a pure condition in Q(C) such that fc < minint(T). Then T is preprocessed for /(i), A;, C if for every v C. k the following holds: If there are a centered family C, a pure condition T" G <5(C") and q e Ai(f) such that C extends C, |C'| = \C\, V < T and (v,T) < ^ then there is p E Ai(f) such that (v, T) < p. LEMMA 2.6.2. Let C be a centered family, f a good Q(C)-name for a real, i,k € UJ, T E Q(C) a pure condition, preprocessed for f(i), k, C. Let C be a centered family extending C, \C'\ = \C\ andT' E Q(C) a pure extension ofT. Then T' is preprocessed for f(i), k, C. PROOF. Let C" be a centered family extending C, \C"\ = \C'\ and T" E Q{C") a pure condition extending T" such that for some 2.6. PREPROCESSED CONDITIONS 41 p G Ai(f), (v,T) < p where v C k. Then C" extends C, |C"| = \C\, T" < T and since T is preprocessed for /(i), /c, C there is q € .4i(/) such that (v, T) < q. However T < T and so (v, V) < q. D REMARK 2.6.3. In particular, if T is preprocessed for f(i), k, C and C is a centered family extending C such that \C'\ = \C\, then T is preprocessed for /(z), k, C. LEMMA 2.6.4. Let C be a centered family, f a good Q(C)-name for a real, i,k € w, T a pure condition in Q(C). Then there is a centered family C extending C, \C'\ = \C\ and a pure condition T" extending T, V e Q(C) such that T is preprocessed for f(i), k, C. PROOF. Let vx,...,vs enumerate the subsets of k. In finitely many steps we will obtain the family C and pure condition T". Consider (vi,T\k). If there is a centered family C[ extending C, \C[\ = \C\ and a pure condition T[ € Q{C[) such that T{ < T\k and for some Pi € 4i(/), (vi,T{) < j^, let Tj = T{ and d = C(. Otherwise let T\ = T, Ci = C. Proceed inductively. At step (s — 1) consider (us,Ts_i) and Cs-\. If there is a centered family C^ extending Cs_i, |C^| = |Cs_i| such that for some pure condition T's £ Q(CS) extending Ts_i, there is ps £ A(/) such that (vs,T^) < ps let T's = Ts and Cs — C's. Otherwise let Ts = Ts_i, Cs = Cs_i. It will be shown that T" = Ts is preprocessed for f(i), k, C = Cs. Let v C k, C" & centered family extending C, \C"\ = \C\, T" a pure condition in Q{C") extending T" and such that for some p £ Ai(f), (v, T") < p. Then v = Vj for some j Es + 1. Since C" extends C", C" 2.7. GENERIC PREPROCESSED CONDITIONS 42 extends C,-_i and furthermore T" < T' < Tj-%. Therefore at stage j we have chosen a centered family Cj and a pure condition Tj G Q(Cj) such that (vj, Tj) < pj for p, G A(/)- But T' < Tj and so (u,, T') < pj. D COROLLARY 2.6.5. Let C be a centered family, T a pure condition in Q(C) and k G ui. Then there is a centered family C extending C, \C'\ = \C\ and a pure condition V e Q{C') extending T, such that for every i < k, T' is preprocessed for f(i), k, C. PROOF. By Lemma 2.6.4 there is a centered family C0 extending C, |C*o| = \C\ and a pure extension T0 € Q{CQ) of T\k, which is preprocessed for /(0), k, CQ. Applying Lemma 2.6.4 at each step, obtain a finite sequence (TJ : % < k) of pure conditions such that Vi € k Ti+i < Ti and a finite sequence of centered families (C; : i T' G Q{C) is an extension of C and since for every i < k V < Ti; by Lemma 2.6.2 for every i < k, T' is preprocessed for f(i), k, C. • 2.7. Generic Preprocessed Conditions LEMMA 2.7.1. Let C be a centered family of pure conditions, f a good Q(C)-name for a real and let T be a pure condition in Q(C). Then there is a centered family C extending C, \C'\ = \C\ and a sequence of pure conditions {Tn : n G uS) C Q(C'), such that (1) T0 (2) Vn G uNi < n, Tn is preprocessed for f{i), n, C. 2.7. GENERIC PREPROCESSED CONDITIONS 43 PROOF. By Lemma 2.6.4 there is a centered family Co extending C, \Co\ = \C\ and a pure extension T0 of T in Q(Co) which is preprocessed for /(0), 0, Co- Proceed inductively. Suppose we have defined Cn, such that \Cn\ = |C„_i|, Tn G Q(Cn) such that for all i < n, Tn is preprocessed for f(i), n, Cn. Then by Corollary 2.6.5 there is a centered family Cn+\ extending Cn, |Cn+i| = |Cn| and a pure condition Tn+i € Q{Cn+i) extending Tn such that for all i < n + 1, Tn+i is preprocessed for f(i), n+1, Cn+i. With this the inductive construction is complete. Then C = U„ea,C„ is a centered family extending C, \C'\ = \C\, which contains the sequence (Tn : n € to). For every n £ u by construction Tn, for all i < n, Tn is preprocessed for f(i), n, Cn. Since C extends C„, \C'\ = \Cn\ by Lemma 2.6.2, for all i < n, Tn is preprocessed for f(i), n, C. • REMARK 2.7.2. The sequence r = (Tn : n € u) is not uniquely determined. Also r is not formally a fusion sequence. Until the end of the section fix a centered family of pure conditions C, a good Q(C)- name / for a real, T € Q(C) and a sequence of pure conditions r = (Tn : n G to) contained in Q(C) which satisfies the conclusion of Lemma 2.7.1 for C, T and /. DEFINITION 2.7.3. Let PT(C, T, f) be the suborder of P(T) consist ing of all finite sequences f = (r0,..., re), I € UJ such that r0 < T0 and for alH : 1 < i < £ and ji = maxint(ri_1), Ti < T^. LEMMA 2.7.4. The following sets are dense in PT(T, C, /): (1) VA; Gw,Efe = {fe Pr(C, T, /) : \f\ > k}, 2.7. GENERIC PREPROCESSED CONDITIONS 44 (2) for all X £ C, n £ u, DT(X,n) = {f£ Pr(C,T,/) : 3rj G f{rj < X and \\rj\\ > n)}. PROOF. Let f = (r* : % E £) be a given sequence in PT(C, T, /). Let jt = maxint(r^_i). Since T^\int(f) and X are compatible, there is a finite logarithmic measure r of level higher than the measure of r^-i and n, which is a common extension of Tje\int(f) and X. Then f^(r) is an extension of r in DT(X, n). • COROLLARY 2.7.5. Let G be a PT(C, T, f)-generic filter. Then (1) RG = UG = (fj : i £ w) is a pure condition of strictly increas ing logarithmic measures such that for every n € u, Rn = (rj : i >n) is a pure extension ofTjn where jn = maxmi(rn_i). (2) In V[G] there is a centered family C' extending C below RG (and so below T) such that \C'\ = \C\. Then in particular for maxx; C". PROOF. By Lemma 2.7.4 RG is a pure condition of strictly increas ing finite logarithmic measures such that Vn € u Rn is a pure extension of Tjn. To obtain the second part note for every X in C and n € ui, the generic filter G meets DT(X, n) and so the sequence Ix = (i : U < X) is infinite. Then RQ A X = fa : i E Ix) is & common extension of RG and X. Furthermore if X Thus C" = {.RG A I : I 6 C} is centered and extends C below J?G; \C'\ = \C\. 2.7. GENERIC PREPROCESSED CONDITIONS 45 ZflG(maxx) = min{j : maxi < minintfo)}. However x C int(i?„) and so maxx < maxint(rfc_i) = jk- Since Rk < Tjk and for every i < jk, Tjk is preprocessed for f(i), jk, C, and so by Lemma 2.6.2 for every * < jk, Rk is preprocessed for f(i), jk, C. However maxx < jk and n < jk- Therefore Rk is preprocessed for f(n), maxs, C. D CHAPTER 3 b = K < 5 = K+ 3.1. Induced Logarithmic Measures In the following K is an uncountable regular cardinal. For complete ness we state MAcountoWe(ft) (see [24]). DEFINITION 3.1.1. MAC(mntaue{K) is the statement: for every count able partial order P and every family V, \V\ < K of dense subsets of P there is a filter G C P such that VD G 2?(G n D + 0). Let .M be the ideal of meager subsets of the real line. Recall that the covering number of Ai, cov(.M) is the minimal size of a family of meager sets which covers the real line. For every regular uncountable cardinal K, cov(A^) > ft if and only if MAcountaue{ti) (see [3]). LEMMA 3.1.2. Let C be a centered family of pure conditions, \C\ < cov(M), f a good Q(C)-name for a real, n G u, T = (U : i G w) pure condition in Q(C) such that for all x G [int(T)]<0J, T\x is preprocessed for f(n), maxx, C. Then the logarithmic measure induced by the family (1) 3i£w such that hi(x PI s;) > 0 where U = (s*, hi) (2) 3w C x3p G An(f) such that (v U w, T\x) < p, takes arbitrarily high values. 46 3.1. INDUCED LOGARITHMIC MEASURES 47 PROOF. TO see that the logarithmic measure induced by the family VV(C, T, f(n)) takes arbitrarily high values, consider an arbitrary finite partition to = AQU- • -UAM-I- By Lemma 2.3.7 there is a pure extension T" of T which is compatible with C and such that int(T') C Aj for some j G M. By \C\ < cov(M) and Corollary 2.5.5 there is a centered family of pure conditions C extending C, \C'\ = \C\ and a pure condition R = {rt : i G to) G Q(C) of finite logarithmic measures of strictly increasing levels, which extends T", and so T. Then / is a Q(C')-name for a real and so An{f) is a maximal antichain in Q{C). Therefore there is a condition (v U w, R') G Q{C) which is a common extension of (v, R) and some q E An(f). By definition of the extension relation there is a finite subsequence (r; : i € [mi, m^) of R, such that w C x = 2 U™ miint(ri). We can assume that rri2 > 1 and so there is i £ [mi, 7712] such that ||rj|| > 0. However R < T and so there isi £ w such that hi(x Pi Sj) > 0. Therefore (1) holds for x. Since R is pure extension of T and T is preprocessed for /(n), maxx, C, there is p G An{f) such that (v U w,T\x) < p and so part (2) holds as well. It remains to observe that x C int(i?) C Aj and so by Lemma 2.1.10 the logarithmic measure induced by Vv(C,T,f(ri)) takes arbitrarily high values. • COROLLARY 3.1.3. Let C be a centered family of pure conditions, \C\ < cov(M), f a good Q(C)-name for a real, n, k G to, T = (U : i G u) a pure condition in Q(C) such that for all finite subsets x of intiT), T\x is preprocessed for f\n), maxcc, C. Then the logarithmic measure induced by the family Vk(C, T, f(n)) of all x G [int(T)] (1) 3i G u> such that hi(si D x) > 0 where U = (SJ, hi), (2) \/v C A;3u; C x3p G A„(/) suc/i tfiat (u U u;, T\x) < p, takes arbitrarily high values. PROOF. Let VQ,..., VL-I enumerate all subsets of k. Then if for all j G L, Xj G VVj(T,f(n)), the set x = x0 U • • - U xL^1 belongs to Vk(C,T,f(n)). To see that the logarithmic measure induced by Vk{C,T, f(n)) takes arbitrarily high values consider an arbitrary finite partition cu = A0U- • -\JAM-I- By Lemma 2.3.7 there is a pure extension T" of T which is compatible with C and such that int(T') C Aj for some j 6 M. By \C\ < cov(M) and Corollary 2.5.5 there is a centered family of pure conditions C" extending C, \C'\ = \C\ and a pure condition R = (rt : i £ to) G Q(C) of finite logarithmic measures of strictly increasing levels, which extends T" and so T. Then in particular for every x G [int(i2)] By Lemma 3.1.2 for every i £ L there is Xi G VVi(C, R, f(n)). It will be shown that x = U{XJ :i£i} belongs to Vk(C,T, f(n)). It is clear that (1) holds for x. To obtain (2) consider any v C k. Then v = Vi for some i £ L. Since Xi(xi C a?) belongs to VVi(C, R, f(n)) there is Wi C ^ and gj G -4.n(/) such that {vi U tOj, R\xi) < qi, and so in particular (vi U ifj, i?\x) < q>j. However R < T, C extends C, \C'\ = \C\ and T is preprocessed for f(n), maxx, C. But then Vt; C Hp 6 Ai(/) such that (v U w,T\x) < p. Therefore x G Vk(C,T,f(n)). It remains to observe that x C int(i?) C A,- and so by Lemma 2.1.10 the logarithmic measure induced by Vk(C, T, f(n)) takes arbitrarily high values. • 3.2. GOOD EXTENSIONS 49 3.2. Good Extensions Until the end of the section let C be a centered family, \C\ < cov(.M), / a good Q(C)-name for a real, T = (£* : i G UJ) a pure DEFINITION 3.2.1. Let P(C,T,/) be the suborder of P(T) of all sequences f = (rj : i G I) such that VWu C iVs C int(r,) which is rj-positive, there are w Q s and p e Ai(f) such that (u U w,T\s) < p. LEMMA 3.2.2. For every k e u the set Ek(C, T, f) = {f€ P(C, T, f) : \f\ >k} is dense in¥(C,TJ). PROOF. Let f = (r0,...,rm_1) be a condition in P(C,T,/) and let ^ = maxint(f). We can assume that m < k. Then ir{() > m and so T\int(f) is an extension of Tm. Then by Corollary 3.1.3 (and \C\ < cov(A^)) the logarithmic measure h induced by Vm = Vm(C,T\mt(f),f(m)) takes arbitrarily high values and so there is x such that h(x) > ||rm_i||. Let rm = (x,h \ V{x)). We claim that an ^(fm) is extension of f which belongs to Em(C, T, /). Let v C m and s C int(rm) = x, h(s) > 0. Then by definition of h there is w C s and p G Am(f) such that (u U w,T\s) < p. In finitely many steps obtain an end-extension of f which belongs to E^. • LEMMA 3.2.3. For even/ X G C, n G u the set Dx,n(C,T,f) = {f G P(C,T, /) : 3r5 G f(^ < X and \\r5\\ > n)} is dense in P(C,T, /). 3.2. GOOD EXTENSIONS 50 PROOF. Let f = (r0,...,rm~i) be a condition in P(C, T, /) and let ^ = maxint(f). Then ir{£) > m and so T\int(f) is an extension of Tm. Furthermore since both X and T\int(f) are in the centered family, there is Y G C which is their common extension. Then in particular Vrr G [int(Y)] It is sufficient to show that f~(rm) belongs to P(C,T, /). Let v C m, s. C x and h(s) > 0. By definition of h there is w C s and a condition q G Am(f) such that (v U io,F\s) < q. Since T is preprocessed for f(m), maxs and C, there is p G Am{f) such that (u U w, T\s) < p. D COROLLARY 3.2.4. Let G be a fitter in P(C, T, /) to/iic/i meets all DXtn(C,T, f) and Ek(C,T, f) for XeC,n,keu;. (1) Then RG = UG = (r» : z G u) is a pure condition of fi nite logarithmic measures of strictly increasing levels such that \li\/v C i\/s C int(ri) which is ri-positive, there is w C s one? p G ^.j(/) suc/i that (v U to, RG\S) < p. (2) Furthermore there is a centered family C extending C below RG (and so below T) such that \C'\ = \C\. PROOF. By Lemmas 3.2.2 and 3.2.3 RG is a pure condition of finite logarithmic measures of strictly increasing levels which is compatible with C. Let % G u, v C i and s C int(rj) which is rrpositive. Then by definition of the partial order, there is w C s and p G Ai(f) such 3.3. MIMICKING THE ALMOST BOUNDING PROPERTY 51 that (v U w, T\s) < p. However RQ < T and so (v U w, RG\S) < P- To obtain part (2) repeat the proof of Corollary 2.5.5 to get a centered family C = {RG A X : X € C} extending C below RQ. D 3.3. Mimicking the Almost Bounding Property u DEFINITION 3.3.1. A family H C u is <*-directed if for every subfamily W such that \H'\ < \H\ there ishGH such that W <* h. THEOREM 3.3.2. Let K be a regular uncountable cardinal, cov(Ai) = K andTi an unbounded, <*-directed family of reals of size K. Let C be a centered family, \C\ < K, f a good Q(C)-name for a real andT € Q(C). Then there is a centered family C, a pure condition R G Q(C) and a real heH such that C C Q(C), \C\ = \C'\, R < T and such that for every centered family C" extending C, for every a 6 [CJ] 00 (a,R)U-Q{c»)3 ieuj(f(i) PROOF. By Corollary 2.7.5 and \C\ < cov(A^), there is a centered family C\ extending C below T, which is of the same cardinality as C and such that there is a pure condition Ti 6 Q(Ci), T\ < T with the property that if T\ = {t\ :iGw) then for every n G ui and every finite subset x of in^JWint^^)), T\\x is preprocessed for /(n), max a;, d. By |Ci| < cav{M) there is a filter G C P(C1,T1,/) meeting C T and D C T for a11 Ek( i, iJ) x,n( i, iJ) k,n E UJ irnd X E Cx. Then by Corollary 3.2.4 the pure condition T2 = UG = (fj : i 6 w) extends Ti, consists of finite logarithmic measures of strictly increasing levels and for all Vz £ uNv C i\/s C int(rj) which is rj-positive, there is w C s 3.3. MIMICKING THE ALMOST BOUNDING PROPERTY 52 and p E Ai(f) such that (v U w, T2\s) < p. For all i E to let g(i) be the maximal k such that there are v C i, w C int(r;), p £ X € C\ let Jx = {i : rt < X} and let Fx be a step function defined as follows: Fx{£) = g(Jx(i + 1)) itt £ e (Jx(i), Jx(i + 1)] where Jx(m) is the m-th element of Jx. Since H is unbounded for all X in Ci there is hx EH such that /ix ^* i^- The cardinality of {hx : X € Ci} does not exceed \C\\ and so is less than K. By the hypothesis on H there is h EH such that hx <* h for every X E C\. We can assume that h is nondecreasing. Note that: (1) VX e CiVi e w(^(i) < Fx(z)) (2) Since 3°°i e w(Fy(i) < hx(i)) and V°°i e u{hx(i) < h(i)) we have 3°°i E u(Fx(i) < h(i)). That is h%* Fx. (3) By part (1) and (2) the set J = {i E LU : g(z) < h(i)} is infinite. (4) Furthermore 3°°i E Jx(Fx{i) < h(i)). Suppose not. Then V°°i E Jx(h(i) < Fx(i)) and so there is mo E to such that V« E Jx if i > mo then (h(i) < Fx(i)). Let m£wbe such that m Jx{fn) — m Jx — ^o- Then a; — Jx(m) = l){(Jx(i), Jx(i + 1)] : i > m} and so if t E ui — Jx(m), then there is i > m such that I E (Jx(i), Jx(i + 1)]. Then /i(£) < h(Jx(i + 1)) < Fx(Jx(i + l)) = J^xW and so h(£) < Fx{£). This implies that h <* Fx which is a contradiction to part 2. (5) However \/i E Jx{Fx(i) = g(i)) and so by part 4 the set Ix = Jx H J is infinite. 3.4. ADDING AN ULTRAFILTER 53 Let R = (r, : i G J). Then for every X G Ci the pure condition #AX = (rj : i € ix) is a common extension of JR and X. Furthermore if X < Y then Ix C Jy since Jx C Jr. Therefore C = {RAX : X 6 d} is a centered family which extends C\ below R which is of the same cardinality as C\. Let C" be an arbitrary centered family which extends (b U io,T2\s) < p for some p 6 A{f)- However R'\s < R\s < T2\s. Therefore (b U w, R'\s) < (b, R') and (b U w, R'\s) < p. Let j G u be such that p lh j = /(z). Then by definition of g we have that j < g(i). Since i G J, g(i) < /i(i) and so (bUw,R'\s) H-Q(C») "/(*) =j < 0(i) < /i(i)". However (b,R') was an arbitrary extension of (a, R) in Q(C"). There fore (a, i?) II-Q(C") "3? G u(i > k A /(i) < /i(i))". Since A; was arbitrary as well (a, i?) \\-Q{c») "3°°i G a/(/(i) < h(i))". D 3.4. Adding an Ultrafilter LEMMA 3.4.1. Let K be a regular uncountable cardinal, cov(M) = K, H C UUJ an unbounded <*-directed family, \H\ = K, VA < K(2A < K). Then there is a centered family C such that \C\ = K, and (1) \\~Q(C) "T~L is unbounded", (2) Q(C) adds a real not split by the ground model reals. 3.4. ADDING AN ULTRAFILTER 54 PROOF. Let J\f = {fa}a Begin with arbitrary pure condition T in Q and Co = {T\v : v G [a>] c T" of V such that int(T") C Ap+1 or int(T") C A 0+1 and T" is compat ible with Cp. By Corollary 2.5.5 there is a centered family C^+1 extend ing Cp below T" such that \C'p+l\ = \Cp+x\. Then by Theorem 3.3.2 there is a centered family Cp+i which extends C'p+1, \Cp+i\ = \C'p+1\ and a pure condition Tg+1 G Q(Cp+\) such that Tp+i < T" (and so in particular iat(Tp+i) C Ap+i or int(T^+i) C Ap+1) and such that for some function hp+\ from the unbounded family TC, for every centered family C" extending Cp+1, If gp+1 is not a good Q(Cp)-name, then by Corollary 2.4.5 there is a centered family C'p+1 extending Cp, \C'p+1\ = \Cp\ such that gp+i is not a (5(C^+1)-name for a real. Let T" G Q(C^+1) be arbitrary. By Lemma 2.3.7 there is a pure condition Tg+1 extending T" which is compatible with C'p+1 and such that int(T^+1) C Ap+i or int(T^+i) C 3.4. ADDING AN ULTRAFILTER 55 -A/3+1- By |C7g+1| < cov(.M) and Corollary 2.5.5 there is a centered family C/3+i extending C'p+l below Tp+1 such that \Cp+x\ = |C*^+1|. If a is a limit let Ca = Up / = U{«MJ),P> :P e Mf),ie coj; e u} where for every i 6 u, Ai(f) is a maximal antichain in Q{C) and so |A(/)| = ^o- For every i e u, p e A(/) let on(p) = min{7 : p G Q(C7)}. Since K is regular uncountable sup{aij(p) : p € v4,(/)} = oti < K and furthermore a = supiew on < K is minimal such that / is U a (5(Ca)-name for a function in UJ. Then / is a name in the list A/". Note that for every /3 > a, f is a <3(C/3)-name and so there is d < K such that / is the name with least index in A/"\{g7+i}7<,5 which is a <3(C,s)-name (note a < S). That is / = 5,5+1. If / is not a good Q(Cs)- name, then we would have chosen the centered family C5+1 such that / is not Q(Cs+i)-name for a real. Then in particular there is i € u and p £ Q(Cs+i) such that p is incompatible with all elements of Ai(f) as conditions in Q(Cg+i)- But then by Lemma 2.2.6 A(/) u {p} is an antichain in Q and so A(/) U {p} remains an antichain in Q(C). Then in particular A(/) is n°t maximal in Q(C), i.e. / is not a Q(C)- name for a real, which is a contradiction. Therefore / = gs+1 is a good 3.5. SOME PRESERVATION THEOREMS 56 <3(C u Va G [u]< (a,Ts+1) lhQ(C) 3°°i € co(f(i) < h5+1(i)). For every 7 G K the set D7+i = {(u,T) e Q(C) : T < T7+1} is dense and so UG C* int(T7+1), which implies that UG is almost contained in c AJ+i or in A 1+1. • 3.5. Some preservation theorems We will use the following well known fact about ccc forcing notions. REMARK 3.5.1. Note that if H a <*-directed family, then for every ccc forcing notion P, (His <* -directed)v . The preservation theorem below will be of importance for the con sistency result to be presented. The proof of Theorem 3.5.2 can be found in Judah and Shelah, [21], Theorem 2.2 (see also [13]). THEOREM 3.5.2. Let H C"u be unbounded such that every count able subset ofH is dominated by an element ofH. //(P7, Q7 : 7 G a) is a finite support iteration, cf(a) = u, such that V7 G a, lhp7 "Q7 is ccc" and lhp7 "H is unbounded'. Then lhpa "H is unbounded'. PROOF. Suppose there is a PQ-generic filter G such that in V[G] there is 3/ G "w dominating H. Let / a P-name for the real /. Let {an}new be increasing and cofinal sequence in a and for every n G u) let 3.5. SOME PRESERVATION THEOREMS 57 fn be a function in V[(JQ„], where Gan = G fl PQre such that for every i G u, fn{i) = j iff 3q G Fa{q \ an G Gan and q \\-a f(i) = j). Then for every n G u there is a function hn G 7i such that V[GaJ t= (hn j£* /„). Since Fa is ccc, there is C G [7^]^ fl V such that {/in : n G w} C C and a function h G Ti fl V(C <* /i). Then in particular for every n & u, there is fc„ such that \/i > kn(hn(i) < h(i)). By assumption V[G] 1= Tt <* f. Then there are p £ G and A; G u such that Vi > k, p Ih /i(z) < f(i). Fix an such that support(p) C an. Then, since V[Gan] \=-hn% fn we have in particular V[GaJ \= 3i > max(kn,k)(fn(i) < hn(i)) and so there is i > max(fcn, k) and condition p' G Gan such that p' lh fn(i) < hn(i) where /„ is a Pan-name for /„. By definition of /„ there is a condition q G Pa such that q \ an G Gan and q \\-a fn{i) = /(»)• Since p \ an, p' and g \ an belong to the generic filter Gan there is q' G Pa which is a common extension of p, p' and g. Then 9' Ih, AW = /(i) < K{?) < Hi) < f(i) which is a contradiction. • LEMMA 3.5.3. Let H be an unbounded family of reals and let C be the Cohen forcing notion. Then I he "H is unbounded'. PROOF. Let / be a W-name for a function in ww. It will be shown that there is h & H such that II- h ^* f. For every p € C let gp(i) = min{jf : 3q < p(q Ih f(i) = j)}. 3.5. SOME PRESERVATION THEOREMS 58 u Then {gp : p 6 C} is countable and so there is g G w D V such that \/p G C(gp <* g). That is for all p G C there is mp G to such that Vi > rnp{gp{i) < 3(0)- Since Ti is unbounded there is h G 7Y which is not dominated by 3, that is the set A = {i G u> : g(i) < h(i)} is infinite. It is sufficient to show that It- 3°°i G A{f{i) < g(i)), since then Ih 3°°i G u(f(i) Let p G C. Suppose there is no q < p such that q Ih 3°°i G -<4(/(i) < g{i)). That is p Ih ^(3°°i G A(/(i) < £(i))) and so p Ih V°°? G A(g(i) < f(i)). That is there ismgw and and extension q of p such that for all i G A, i > m, q Ih g(i) < f(i). Let ieibe greater than m and mq. Let 5' be an extension of q and let j G a; such that COROLLARY 3.5.4. Let H Q^LO be unbounded and let C(«) 6e i/ie forcing notion for adding K Cohen reals. Then (TC is unbounded)v " . PROOF. By Lemma 3.5.3, Theorem 3.5.2 and Lemma 3.5.6. • The proof of Lemma 3.5.5 can be found in Bartoszynski, Judah, [4]. LEMMA 3.5.5. Let Ti C uu be an unbounded, <*-directed family, \7i\ = K, P a forcing notion, |P| < K. Then (H is unbounded^ . PROOF. Let / be a P-name for a function in "to. For every p G P and i G u let gp(i) — min{j : 3q < p(q Ih f(i) = j)}. Since 1 (H is unbounded) '' for every pGP there is a function hp G H D V which is not dominated by gp. However \{hp : p G P}| < n and so there 3.5. SOME PRESERVATION THEOREMS 59 is h G H fl V which dominates all hps. That is for every p6P there is np G UJ such that \fi > np(hp(i) < h(i)). Suppose p £ F such that p Ih "H <* /". Then there is p0 < p and n0 G UJ such that \/i > no, po Ih h(i) < f(i). Let i > max{n0, np} be such that gpo (i) < hpo (i) and let q be an extension of po such that Q H- 9p0(i) = /(»)• Then q Ih «/(i) = gpo(i) < hPQ{i) < h(i) < f(i}" which is a contradiction. • The last two Lemmas in this section summarize some well known facts of finite support iterations of ccc forcing notions. LEMMA 3.5.6. Let K be an ordinal of uncountable cofinality and let {Fa, Qa '• ot < K) be a finite support iteration of ccc forcing notions. Then every real in V¥K is obtained at some initial stage of the iteration of countable cofinality. That is p w ¥a "OJ n V " = U{ o; n V : a < K, cf(a) = u}. PROOF. Let / be a PK-name for a real. We can assume that / is of the form / = U{((«, j*),p) : p G A^i G oj,jp G a;} where each Ai is a maximal antichain of conditions in PK deciding f(i). Since FK is ccc every Ai is countable. Furthermore F is a finite support iteration and so in particular for every p G Ai, the support of p is finite. Therefore for all i G to, oti = sup{a^ : p G Ai} where af = max support(p) is of countable cofinality and so is smaller than K. But then a = supigw a^ is also of countable cofinality (thus a < K) and / is a Pa-name. • 3.6. b = K LEMMA 3.5.7. Let K be a regular uncountable cardinal. Let (Pa, Qa : a < K) be a finite support iteration of ccc forcing notions of length K. U Let G be a ¥K-generic filter and let A C V[G] D UJ, \A\ < K. Then A is obtained at some proper initial stage of the iteration. PROOF. For every / in A let / be a PK-name for /. By Lemma 3.5.6 for every / there is an ordinal aj of countable cofinality such that / is a Pa/-name for a real. Let a = sup{a/ : / E A}. Then cf(a) < \A\ < K and so a < K. It remains to observe that A is contained in V[Ga] where Ga = GnIPQ. D 3.6. b = K < s = K+ DEFINITION 3.6.1 (Hechler, [20]). Let A be an infinite set of func tions in WLQ. Then M(A) is the forcing notion consisting of all pairs n <0J (s,F) where s e Un£ul uj and F e [A] with extension relation de fined as follows. We say that (slyFi) extends (s2,F2) (and denote this by(s1,F1)<(S2,F2))if (1) s2 C Sl) F2 C Fu (2) V/ G F2Vk E dom(si)\dom(s2) we have si(k) > f(k). The following is a well known fact about Hechler forcing, see [20]. LEMMA 3.6.2. Let A be an infinite set of functions in "u. Then the partial order M(A) is a-centered, adds a real dominating A and is of the same cardinality as the set A. PROOF. Note that if (si,F{) and (s2, F2) are elements of M(A) such that si = s2 = s, then (s, FiUF2) is their common extension. To obtain 3.6. b = ft \s\ + 1 we have f(i) < /G^)- d THEOREM 3.6.3 (GCH). Let K be a regular uncountable cardinal. Then there is a ccc generic extension in which b = K < s = K+ . PROOF. Obtain a model V of b = c = K by adding K-many Hechler reals (i.e. do a finite support iteration of length K of Hechler forcing, see [19] and [8]). Let TL — V D^LO. Then 7i is unbounded and every subfamily of 7i of cardinality less than K is dominated by an element of H. Furthermore in V for every A < K, 2A = K. By transfinite induction of length K+ define a finite support iteration of ccc forcing + notions {(Pa : a < K ), (Qa : a < K)) as follows. Suppose a is a limit and for every /? < a we have defined a ccc forcing notion P^ and a P^-name Qp such that in VPf} the family 7i is unbounded and Ihp^ "Q^ is ccc ". Let Fa be the finite support iteration of (FfrQp: P (1) By Theorem 3.5.2 the family Ji remains unbounded in VPa. ¥a (2) Since PQ is ccc, by Remark 3.5.1 H is <*-directed in V . (3) In V¥a for every A < K, 2A < K. If a is a successor, a = (3 + 1 and P^-has been defined, then: (1) Let Qp be a Fp name for C(K), i.e. the forcing notion for adding « Cohen reals and let P^+i = Fa = Fp * Qp. Then 3.6. b = K < s = K+ 62 (a) By Corollary 3.5.4 the family 7i is unbounded in VPa. Fa (b) Since 1PQ is ccc, by Remark 3.5.1 Ji is <*-directed in V . (c) Forcing notions with the countable chain condition do not collapse cardinals and so VA < K(2A < K). (d) In V¥a the covering number of the meager ideal M. is K. (2) Therefore in VFa the hypothesis of Lemma 3.4.1 holds and so there is a centered family of pure conditions C such that Q{C) preserves H unbounded and adds a real not spit by [u>]w n VPa. Let Qa be a Pa-name for Q(C) and PQ+1 = Fa * Qa. Then: (a) By part (2) of Lemma 3.4.1, Ji is unbounded in VFa+1. Po,+1 (b) Since Pa+i is ccc, H remains <*-directed in V . X (c) Also by the ccc of PQ+1 VA < K(2 < K). (3) In V¥a+l let A C WUJ be an unbounded family, \A\ < K and let Qa+i be a Pa+rname for M(^l). Let PQ+2 = Pa+i * Qa+i- (a) Then since |H(^4)| = \A\ < n, by Lemma 3.5.5 the family H remains unbounded in V¥a+2. (b) Since M(A) is ccc, by Remark 3.5.1 every subfamily of 7i of size less than K is dominated by an element of 7i. Fa 2 X (c) Again by the ccc of PQ+2, in V + for all A < K(2 < K). (d) Furthermore A is bounded in VFa+2. With this the inductive construction is complete. Let P = PK+ be + + the finite support iteration ((PQ : a < K ), (Qa : a < K ))- Then P is a ccc forcing notion and in VF we have that 2W = K+. Let A be a subfamily of [u/jTiy1'' of cardinality less than K+. Then by Lemma 3.5.7 + there is a < K such that A C V[Ga] where Ga = G n PQ and G is 3.6. b = K < s = K+ 63 a P-generic filter over V. Then by the inductive construction of P, in ¥ + V[GQ+3] there is a real which is not split by A. Therefore V N s = re . By Theorem 3.5.2 and the construction of P the family TC is unbounded in Vp. Since every family of reals in V¥ of size less than K is obtained at some initial stage of the iteration, using a suitable bookkeeping device (by step (3) of the successor case in the inductive construction of P) one can guarantee that any such subfamily is bounded in Vr and so VF t= b = K. • REMARK 3.6.4. Alternatively in the model V defined in the proof of Theorem 3.6.3 one can define a finite support iterated forcing con + + + struction {(PQ, : a < re ), (Qa : a < K )) such that for every a < re , IHp„ "Q« is ccc and |QQ| = c" as follows. If a is a limit and P^Q/j have been defined for every /? < a let Pa be the finite support iteration of (P/j,Q/3 : (3 < a). If a = /3 + 1 and P^ has been defined, then let Vg = V^0 and let Hi be the forcing notion for adding n Cohen reals. Then in Vp 1 by Lemma 3.5.4, H is x 1 unbounded and VA < n(2 < K), COV(M) = K. Therefore in Vf the hypothesis of Lemma 3.4.1 hold and so there is a centered family of pure conditions C such that Q(C) adds a real not split by V^1 D [w]w (and so not split by VpD[wf) and preserves H unbounded. Then let H2 be a Hx-name for Q{C) and in vf1 (Hj * H2) * H3, and let Fa = F0 * Q^g. CHAPTER 4 Symmetry 4.1. Q(C) which preserves unboundedness Suppose for every unbounded family of reals TL C UUJ there is a centered family of pure conditions C = Cu in the partial order Q such that Q(C) adds a real not split by the ground model reals and at the same time preserves Ji unbounded. Then let V be a model of GCH and Vi a generic extension obtained by adding K Hechler reals H (for K- regular uncountable cardinal). Proceed with a finite support iteration (Qa : a < A) of length A over V\ where (1) for every even a, Qa = Q(Ca) for Ca a centered family of pure conditions such that Q(Ca) preserves TC unbounded and adds a real not split by the ground model reals, and (2) for every odd a, Qa = H^a) is the Hechler forcing associated with a family of reals Aa obtained at a previous stage of the iteration which is of cardinality less than K. Then V®x would satisfy b = K < s = A. However there are certain difficulties in obtaining such centered family of pure conditions. One may try to proceed along the lines of Theorem 3.3.2, dropping the requirement that \C\ < \H\. Then it would be sufficient to guarantee that for every X G C\ the intersection Ix = Jx H J is infinite, where C\, Jx, J are defined as in the proof of Theorem 3.3.2. For this it 64 4.1. Q{C) WHICH PRESERVES UNBOUNDEDNESS 65 would be sufficient to provide a filter G in P = P(Ci,Ti, /) meeting Dj(X,n) = {f 6 P : 3n G f s.t. i E J,Ti< X and ||ri|| > n} for all X 6 Ci, n G a>. It is not difficult to show that for all A G [u;]w, X G Ci the corresponding set DA(X, n) is dense in P(Ci, Ti, /). There fore the existence of such a filter would require meeting continuum many dense sets which is not possible. One way to sidestep this diffi culty is exactly what is done in Theorem 3.3.2, namely to require that every subfamily of Ti of cardinality smaller than \H\ is dominated by an element of H and to consider only centered families of cardinality less than the cardinality of the unbounded family H. As is established in chapter III this leads to the consistency of b = K < s = K+ for arbitrary regular uncountable K. Observe that the restrictions on the unbounded family H. as well as the centered family C, prevent further iteration and so also further generalization of the same construction. Another way to sidestep the difficulty in preserving a small un bounded family unbounded, is to consider generic centered families, that is centered families of names for pure conditions (see Theorem 5.4.1) Let T be a set of ordinals. Then C(r) denotes the forcing notion of all partial functions from rxwtow with extension relation reverse inclu sion. That is p < q if q C p and so in particular C(co2) is the forcing notion for adding u2 Cohen reals. In the last three chapters we will examine the existence of a countably closed forcing notion which has the K2-chain condition and which adds a centered family C of C(a>2)- names for pure conditions, such that Q(C) preserves the first u>i Cohen 4.1. Q(C) WHICH PRESERVES UNBOUNDEDNESS 66 reals unbounded and adds a real not split by VC^J2'1 n [u]". Consider the following partial order. DEFINITION 4.1.1. Let P' be the forcing notion of all pairs p = (Tp, Cp) where Tp is a countable subset of u2 and Cp is a countable centered family of C(rp)-names for pure conditions with extension re lation defined as follows. For every p and q in P' let p < q if Tq C Tp andlHc(rp)"CgCQ(Cp)". Then in particular P' is countably closed. Note that if G is V- generic then CQ = U{Cp : p £ G} is centered family of C(a;2)-names for pure conditions. Let GQ be C(w2)-generic filter. Then it would be sufficient to guarantee that UH = U{u : 3X s.t. (u, X) e H} where H is <3(CG)-generic over V[Go][G] is not split by c{U2) r yc(w2) n ^ = v * n [uf. This amounts to obtaining the following Lemma: LEMMA 4.1.2. Let T be a countable subset of 102, C a countable centered family of OY)-names for pure conditions. Let A be a C(r)- name for an infinite subset of to. Let G be a C(T)-generic filter. Then in V[G] there is a pure condition X such that int(X) C A or int(X) C Ac and a countable centered family C' extending C below X. The second task is to preserve the collection of the first UJI Cohen reals unbounded. Note that equivalently we might aim in preserving 4.1. Q(C) WHICH PRESERVES UNBOUNDEDNESS 67 unbounded any subfamily of the Cohen reals of size ui. Let / be a C^) * <5(C,G)-name for a real in V[G]. Then there is a countable subset T of u)2 and a countable centered family C C CQ of C(r)-names for pure conditions such that / is a C(r) * Q(C)-name for a real. Then it would be sufficient to show the following. LEMMA 4.1.3. Let f be a C(rp) * Q(Cp)-name for a real, let S £ u>i\Tp and let h = UG$ where G$ is the C({5})~canonical name for the generic filter. Then there is a countable centered family C' of C(r U {5})-names for pure conditions extending C such that for every centered family C" of C(u>2)-names for pure condition which extends C', lhc(w2)*Q(C») "h ii* /" • Observe that if / is a C(r) * <5(C)-name for a real, where T is a countable subset of o>2, C is a countable centered family of C(F)-names for pure conditions, then for every I"" € [^2]^, such that T C V, f is also a C(r') * <3(C)-name for a real. However if G is a centered family of C(r')-names for pure conditions extending C, that is Ihc(r') G C Q(C') then it is not necessarily the case that / is a C(r') * Q(C')-n&me for a real. Lemma 4.1.3 holds, as it will be shown later, for names / which are good in the sense that whenever C" is as above, / is also a C(r') * Q(C')-n&me. An important point in preservation of the first o^ Cohen reals is the fact that for every C(UJI) * <3(CG)-name for a real /, there is a € G such that / is a good C(ra) * Q(Ca)-name for a real (see discussion following Definition 5.4.2). The main difficulty in realizing this project is the K2-ehain condi tion. Work in a model of CH and consider a collection {pa : a & 1} 4.1. Q{C) WHICH PRESERVES UNBOUNDEDNESS 68 of ^2-many elements of P'. For every a G I let Ta = FPa, Ca = CPa. By CH and passing to a subset we can assume that for all a, (3 G / the order types of Ta and T^ are the same. Furthermore by the Delta System Lemma we can choose a subfamily {pa : a G J} for some J C 7, \J\ = ti2 such that for all a, (3 G J, Ta C\ Tp = A, sup A < minra\A and for all a < (3 in J, suprQ\A < minT^A. Also we can assume that there is an isomorphism ia^ :Ta = Yp such that ia:p f A = id and {ia^(X) : X G Ca} = Cp. Therefore it is sufficient to obtain: LEMMA 4.1.4. Let p,q be conditions in F' such that A = Tp n Tq, sup A < minrp\A < supTp\A < minrg\A and there is an isomor phism i : Tp ~ Tq, such that i \ A = id and Cq = {i(X) : X G Cp}. Then there is r G P' such that r < p and r < q. The main argument of Lemma 4.1.4 is the claim below. LEMMA 4.1.5. Let X G Cp. Then there is a C(rp U Tq)-name for a pure condition X such that IHc(rpur9) X < X andX < i(X). Indeed if 4.1.5 holds, then r = (Tr,Cr) where Tr = Tp U Tq and Cr = Cp U Cq U {Xx : X G Cp) where for every X G Cp, Xx is C(rp U rg)-name for a pure condition extending X and i(X) would be a common extension of p and q. In order to guarantee Lemma 4.1.5, we have to impose certain combinatorial property on the names for pure conditions (see Definition 4.3.2 and Definition 6.1.3). We refer to names that have this property as symmetric since one of its defining characteristics is that different evaluations of the name are compatible 4.2. SYMMETRIC NAMES FOR SETS OF INTEGERS 69 pure conditions (see Lemma 4.5.1). The same combinatorial property can be imposed on names for infinite sets of integers (Definition 4.2.2). What might be considered of independent interest is the fact that in every Cohen generic extension the collection of subsets of LO which do not have symmetric names forms an ideal (see Corollary 4.2.10). Fur thermore we have to accomplish the entire construction, in particular define a partial order analogous to 4.1.1 and obtain statements anal ogous to Lemmas 4.1.2, 4.1.3, 4.1.4 and 4.1.5 remaining within the class of names for pure conditions which have the given combinato rial property. In chapters IV and V we develop a particular case of this combinatorial property, which completes the fr^-chain condition in case that the root of the Delta system is empty and establish the construction within the class of names for pure conditions which have this property - see Lemma 4.4.6, Theorem 5.4.1 and Lemma 5.4.3. In the last chapter we give a generalization of this combinatorial property (Definitions 6.1.3 and 6.1.1) and demonstrate the chain condition for non-empty root (Lemma 6.2.2). 4.2. Symmetric Names for Sets of Integers DEFINITION 4.2.1. Let X be a Cohen name for an infinite subset of u. Then for every p G C let hullpX = {j : 3q < p(q lh j G X)}. DEFINITION 4.2.2. A Cohen name X for an infinite subset of a; is said to be symmetric if for every finite number of conditions pi,...,pkia C and every M G OJ, there is m > M and extensions pi < pi:... ,pn < pn such that for every i < k, pi lh fh G X. 4.2. SYMMETRIC NAMES FOR SETS OF INTEGERS 70 LEMMA 4.2.3. Let X be a Cohen name for an infinite subset of u. Then X is symmetric if and only if for every finite number of conditions Pi,... ,pn inC the set P|i=1 hullp^X) is infinite. EXAMPLE 4.2.4. Every check name for an infinite subset of u is symmetric. LEMMA 4.2.5. The Cohen generic real has a symmetric name. That is if G is the canonical name for the generic filter, then X = [J G is a symmetric name. PROOF. Let p\,... ,pk be a finite number of conditions and nGw. Then there is j > n which does not belong to the domain of of the given conditions. Then for all £ = 1,..., k, qt = pe U {(j, 1)} extends of pe and qe Ih j G X. That is j G f\Li hullK(X). • In the remainder of this section it will be shown that in the Cohen extension the family of subsets of LO which do not have symmetric names forms an ideal. LEMMA 4.2.6. Suppose X andY are C-names for subsets ofu such that \\- X C.Y and Y is not symmetric. Then X is not symmetric. PROOF. Suppose X is symmetric. Since Y is not symmetric there are conditions p\,..., pn in C such that f)™=1 hullPi (Y) C M for some M G to. Since X is symmetric there are extensions pt < pi and m> M such that for every i < n, pi II- rh G X. Then for every i < n pi Ih fh G Y and so m G PlLi^W-^") which is a contradiction. • 4.2. SYMMETRIC NAMES FOR SETS OF INTEGERS 71 DEFINITION 4.2.7. A Cohen name X is symmetric below a condi tion p if for every finite family pi,... ,pn of extensions of p and M € OJ there are extensions pi < Pi for all i and m > M such that ft lh m G I. LEMMA 4.2.8. Let X be a symmetric name for a subset of u and let Y, Z be Cohen names such that lh X = Y U Z. Then for every p G C either there is q < p such that Y is symmetric below q or there is q PROOF. Suppose not. That is there is p G C such that for every q < P, Y and Z are not symmetric below q. Then in particular there are extensions pi,... ,Pk of p and n0 G to such that f]i=1 hullPi(y) C n0. Similarly for every i there are extensions qt. < Pi fovj — 1,..., ki and 3 Hi £ u such that f]i =1h.ullqi (Z) C n^. Let M = maxj REMARK 4.2.9. Whenever X is a P-name for a pure condition and p e P, let X \ p = {(£, q) : q COROLLARY 4.2.10. Let G ba a Cohen generic filter. Then the collection Tnsym € V[G] of subsets of u which do not have symmetric names forms an ideal. PROOF. Let lh X = Y U Z where X is a symmetric name for an infinite subset of ui. Then the set D of all conditions p € C such that Y is symmetric below p or Z is symmetric below p is dense. Let E be a maximal antichain contained in D and for every e £ E define X* \ e = Y f e if y is symmetric below e and let X* \ e = Z f e if Z is symmetric below e. Then X* is a a Cohen name for an infinite subset of cu such that lh (X* C X) A (X* = Y V X* = Z). With every e € E associate a symmetric name X* for an infinite subset of w as follows. Let X* \ e = X* \ e. Let e' be a condition in E distinct + + from e. There is an isomorphism ieei : C (e) —*• C (e') where for every p 6 C, C+(p) = {q E C : q < p}. Then for every e' £ E\{e} let XI \e> = iee,(X: \e). Let G be a Cohen generic filter, X = X[G\, Y = Y[G] and Z = Z\G\. Then G D E = {e} and so V[G] \= X*[G] = Y or X*[G] = Z depending on whether Y or Z is symmetric below e. Thus either Y or Z has a symmetric name. • REMARK 4.2.11. Xnsym does not contain infinite subsets from the ground model V, since every check name for an infinite subset of u is symmetric. Note also that a symmetric name is necessarily a name for an infinite subset of u and so every finite subset of to belongs to Xnsym. 4.3. SYMMETRIC NAMES FOR PURE CONDITIONS 73 4.3. Symmetric Names for Pure Conditions In the following LM denotes the family of all finite logarithmic measures. For every n E u let Ln be the set of all finite logarith mic measures x such that \\x\\ > n and minint(x) > n. Just as in Section 4.2 we can give the following definition: DEFINITION 4.3.1. Let Ibea Cohen name for a pure condition. Then for every p E C let hullp(X) = {xeLM : 3q(q < p)(q lh x < X)}. DEFINITION 4.3.2. Let X be a C-name for a pure condition. We say that X is symmetric if for every n E u and every finite number of conditions pi,...,pk there is x E Ln and extensions p\ < p\,..., pk < Pk such that for every £ = 1,..., k (pi lh x < X). DEFINITION 4.3.3. A name for a pure condition X is symmetric below a given condition p if for every M 6 u and finite number of extensions pi,..., pn of p there are extensions p\ < p\,..., pn < pn and a measure x E LM such that for every I = 1,..., n pi lh "x < X". PROPOSITION 4.3.4. LetX be a Cohen name for a pure condition. The following are equivalent: (1) X is symmetric. (2) For every finite number of extension Pi,. • • ,Pk of p and n E u> the intersection (f]i=1 hullp^X)) D Ln is nonempty. (3) For every finite number of extensions p\,..., pk of p the set fli=i hullp^X) contains a pure condition. 4.3. SYMMETRIC NAMES FOR PURE CONDITIONS 74 PROOF. Part (2) is just a reformulation of part (1). Assume (1) and let X be a symmetric name for a pure condition. Let pi,...,pk be some finite number of conditions and new. Then there is xn 6 Ln and extensions phn < Pi,..-,Pk,n < Pk such that hull Pi>n lh xn < X (VZ < k) and so xn G nti K PO PI Ln- Then for A;„ = max{|| int(x„)} there is xn+i G Lkn, and extensions pi)n+i < Pi,--- ,Pk,n+i < Pfe such that pe,n+i ^ xn+1 < X (W < A;) and so in n particular xn+1 belongs to Hi=i ullPi (X). Proceeding inductively we can choose a pure condition (xn : n £ cu) contained in f\=i hullPi(X). To see that (3) implies (1) fix any p\,... ,pk finite number of con ditions and let n € u. By assumption f]i=1 hullPi (X) contains a pure condition (xt : i G u) = R of logarithmic measures of strictly in creasing hight. However xn G Ln f](f]i=1hu\lPi(X)) and so for some Pi REMARK 4.3.5. Thus a C-name for a pure condition is not sym metric iff there are conditions pi,...,pk and M G oo such that (njL1huliK(X))nxM = 0. REMARK 4.3.6. If a finite logarithmic measure x does not belong to n^=1hullPi(X) then there is an index i < k such that pi lh x ^ X. LEMMA 4.3.7. Let X and Y be C-names for pure conditions such that lh X < Y. IfY is not symmetric, then X is not symmetric. PROOF. Suppose that X is symmetric, but Y is not symmetric. Then there are conditions pi,...,pk such that Di=i hmW^0 does not 4.4. AN ULTRAFILTER OF SYMMETRIC NAMES 75 contain measures of level greater than M for some M E to. Let j > M. Since X is symmetric there are extensions qi < pi,-- • ,qk < Pk and x G LM such that for every i < k, qt lh x < X. But then qe\\- x 4.4. An ultrafilter of Symmetric Names DEFINITION 4.4.1. The finite logarithmic measure x is said to be stronger than the finite logarithmic measure y if x is of measure greater than the measure of y and minint(x) > maxint(y). We will denote the fact that x is stronger than y with x > y. REMARK 4.4.2. Whenever p and q are incompatible conditions we will denote this by pl.q. LEMMA 4.4.3. Let X be a symmetric Cohen name for a pure con dition and let A be a name for an infinite subset of'u. Then for every Pi,... ,Pk in C and every M 6 UJ there is a finite logarithmic measure z G LM and extensions p\ < pi,... ,pk < Pk such that \/i < n, pi lh "z k PROOF. Let so = 2 +M. Since X is symmetric there are extensions Pi,i < Pi, • • • ,Pi,k < Pk and x G LSo such that for every i < k, p^i lh x < X. Let s = max a; +1. Extend p\^ to a condition p2,» such that for some djCs, p2,» lh (^4 \ s) = dj and so in particular if bt = s\aj then c P2ii ih (A r 5) = K In the ground model we can partition x into 2fc subsets {^ : j < 2fc} k such that Vj < 2 \/i < k Zj is contained in at or &,. Furthermore by 4.4. AN ULTRAFILTER OF SYMMETRIC NAMES 76 k Lemma 2.1.3 there is jo < 2 such that the measure of z = Zj0 is at least M. Then for every i < k we have c p2,i lh "(z < X and z C i) or (i < X and 5 C i )". Then for every i < k there is a further extension pi < p2j such that a c pi lh "i < X and i C A" or pt lh z < X and z C i ". D LEMMA 4.4.4. Z,e£ X be a C-symmetric name for a pure condition and A a C-name for an infinite subset of u. Then there is a Cohen symmetric name for a pure condition Y such that lh Y < X and Vi 6w lh uint(Y(i)) C A orint(Y(i)) C ic". PROOF. Fix an enumeration {pn : n G LO} of C Find an extension Pofl of p0 and a finite measure xo such that po,o lh "xo C A" or "po,o lh x0 < X A int(x0) C ,4 ". Let ^0 = {a0,s : s € w} be a maximal antichain in C — C(po,o) such that for every s & u there is u a measure XQ,S such that ao,s lh xo,s < X A int(xo,s) C A" or a0)5 lh c "^o,s < X A int(x0,s) C .A ". Let RQ = {(p0,o,xQ)} U {(a0,s,x0,s} : s G u;}. Proceed inductively. Suppose we have defined conditions {Pn-i,e}een and a finite logarithmic measure xn_a such that for every £ C pn-i,e lh "x„_i < X A int(x„_i) C J4 ". Furthermore suppose we have defined a maximal antichain An-i = {an-i,s '• s € UJ} in C — C({p„_i/}^6„) such that for every s G u> there is a finite logarithmic measure xn-i,s such that an_i)S lh "xn_iiS < X A int(xn_i)S) C A" or 4.4. AN ULTRAFILTER OF SYMMETRIC NAMES 77 u 0*1-1,3 ll~ %n-i,s < X A int(x„_liS) C A" and finally we have defined Rn-l = {(Xn-l,Pn-lte)}(£n U {(£ „_i,s, a„-l,s) : S G w}. If p„±{p„_iij}i<„_i then there is a„_i)S G Ai-i compatible with pn and we can fix a common extension bn and let yn = £„-i,s- Other wise let bn be a common extension of pn and some pn-ij for j G n and let y„ = a;„_i. By the Lemma 4.4.3 there are extensions p„]0 < Pn-i,o, • • • ,Pn,n-i < Pn-i,n-i,Pn,n < ^n and a finite measure xn stronger than a;„_i and yn such that \/i c (Pn,i l\- xn Fix a maximal antichain An = (a„]S : s 6 w) in C - C({pnii}j 5 (1) 3i G UJ such that a„]S < an-i^s (2) 3x„,s measure stronger than xn-i^ such that - fln.s II "^n,5 < -X A int(s„)S) C A" or an>s Ih "in)S Let Rn = {{Pn,t)^n)}j REMARK 4.4.5. Note that Ri is a name for the i-th. measure of Y. LEMMA 4.4.6. Let G be a Cohen generic filter. In V[G] let X be a pure condition with symmetric name X and let A EV[G] be an infinite subset of UJ. Then in V[G] there is a pure condition Z extending X, which has a symmetric name and such that int(Z) C A or int(Z) C Ac. 4.4. AN ULTRAFILTER OF SYMMETRIC NAMES 78 PROOF. Let Y be the name constructed in Lemma 4.4.4. Then there are C names R and T such that lh R = (Y(i) : Y(i) C A) and lh f = (Y(i) : Y{i) C Ac). Then II- .RUT = Y. We claim that for every p G C there is an extension g < p such that R is symmetric below q or T is symmetric below q. Suppose not and let p be a condition which does not have an exten sion with the desired properties. Then there are extensions Pi,---,Pk of p such that for some no, (fl»=i hullPi(i?) P| Lno = 0 and respectively + for every i < k there are {Qij}/=i Q C (pi) such that for some ni; (P| 4j hullqij(T)) f] Lni = 0. By construction of Y there are extensions an q~%,j < Qij d a measure a; of level higher than {nQ,..., n^} such that q~itj \\- x GY. Then for all i,j qij \\- x E RUT and so there is a further extension ttj < qij such that i,j lh x € R or ijj lh x € T\ If for every i < k there is some j < ti such that ij^ lh x E R, then since Uj < Pi we obtain that x is in f]i=1 hullft (i?) which is a contradiction since x is of measure greater than n0. Otherwise, there is some i < k such that Vj = 1,... ,£, ttj lh x £ T. But then x is in fill nun9i fi which is a contradiction since the measure of x is greater than rij. Therefore the set D of all p € C such that R or T is symmetric below p is dense in C. Fix a maximal antichain i? contained in D. Define Y* as follows: for every e € E let F* f e = i? |" e if R is symmetric below e and let Y* \ e = T \ e otherwise. Then lh (Y* Furthermore for every e € E let Ye be a symmetric name defined as follows. Let Ye f e = R \ e if R is symmetric below e and let Ye \ e' = + iee'(Ye T e) for every e' G £'\{e}, where iee> is an isomorphism of C (e) and C+(e'). Note that for every e £ E, \\- Ye = R\/Ye = f. Now, since G is Cohen generic there is e £ G H E. Then Ye is a symmetric name for JR[G] or T[G], and so in particular for an extension of X the underlying infinite set of which is contained in A or in Ac. • As a straightforward generalization of the above one obtains: COROLLARY 4.4.7. Let G be a Cohen generic filter. If, in V[G] X is a pure condition with a symmetric name and A0 U • • • U ^4n_i is a partition of u into finitely many sets, then there is a pure condition Y extending X which has a symmetric name and such that int(Y) C Aj for some j € OJ. In particular we obtain the following result: COROLLARY 4.4.8. Let G be a Cohen generic filter and let X be a pure condition in V[G] with symmetric name X. Let A € V[G] D [u]u which does not have a symmetric name. Then in V[G] there is a pure condition Y extending X such that int(Y) C Ac. PROOF. By Lemma 4.4.6 there is a symmetric name Y for a pure condition such that in V[G], Y[G] = Y < X and int(F) C A or int(F) C Ac. Suppose V[G] 1= int(y) C A. Let A be a Cohen name for A and let p £ G be a condition in G such that p lh int(F) CA If A is symmetric below p then the set A does have a symmetric name, which 4.4. AN ULTRAFILTER OF SYMMETRIC NAMES 80 is a contradiction to the hypothesis of the lemma. Therefore there is a finite set of extensions of p, Pi,-..,Pk such that f]i=1 h.ul\Pi(A) C M for some M & UJ. However Y is symmetric and so there is x G LM and extensions qi < Pi such that for every i <& lh x < Y. Then for every i, qi \\- x Q A which implies that int(x) C f]i=1 hullK (A). This is a contradiction, since x e LM implies that minint(a;) > M. D In his original work from 1984 S. Shelah works with a restriction of the partial order Q to a suborder Q[I], where / is an ideal on V{ui) containing all finite subsets, which consists of all conditions (a,T) in Q, where T = (^ : i £ u), having the property that for every A in I the sequence T D Ac = (ij : int(ij) 0 A = 0} is a pure condition. If G is a Cohen generic filter, in V[G] the collection InSym of subsets of LJ which do not have symmetric names forms an ideal (containing all finite subsets of a;) and so in V[G] we can consider the analogous partial order Qs[Insym] where Qs is the suborder of Q consisting of conditions with symmetric name for the pure part. COROLLARY 4.4.9. Let G be a Cohen generic filter. Then V[G] f= (Qs = QslLsym})- PROOF. In V[G] let X = (xi : i € a;} be a pure condition with symmetric name X and let A be a subset of u which does not have a symmetric name. Let Z be a name for Z = (xt : mt(xi) C\ A = 0). By Lemma 4.4.8 there is a pure condition Y < X which has a symmetric name Y such that int(Y) C A. Then Y < Z and so there is a condition 4.5. EXTENDING DIFFERENT EVALUATIONS 81 p £ G such that p\\-Y < Z. By Lemma 4.3.7 Z is symmetric below p, which implies that Z has a symmetric name. • Whenever X is a P-name for an infinite subset of UJ (resp. a P- name for a pure condition) where P is a forcing notion, such that for every finite set of conditions Pi,- • • ,pn in P and integer M there are extensions pi < p1}... ,pn < pn and m > M (resp. a finite logarithmic measure x G LM) such that for all j = 1,... ,n pj II- rh G X (resp. pj lh x < X) we will say that the name X is symmetric. Also if we want to emphasize that X is a P-name, we will say that X is P-symmetric. 4.5. Extending Different Evaluations LEMMA 4.5.1. Let X be a Cohen symmetric name for a pure con dition. Let C„ = C x • • • x C be the product of n-copies of C Then there is a Cn-symmetric name for a pure condition X such that for all j Cn-generic filters G, for all j = 1,... ,n V[G] N X[G] < X[G ], where Gi is the j-th projection of G. PROOF. Let {pTO}mew be an enumeration of C„. Then for every l T m € cu, pm = (p m,... ,p ^n) where p>m £ C Consider px. Since X is C-symmetric name there are extensions p\ 1 < p\,..., p\ x < p\ and a 3 finite logarithmic measure xx such that for every j = l,...,n,p 11 lh xi < X. Let piti = ip\i,- • • ,Pii) and let R[ = {(pitl,Xi)}. Fix a maximal antichain of conditions Ai = {a\iS : s € to} in C„ — Cn(p\,i) such that Vs£w, there is a finite logarithmic measure xljS such that for every j = 1,... ,n, a{s lh ii,s < X. Let R" = {(ai)5, ix>s) : s G ui} and let Rx = R[ U i?'/. 4.5. EXTENDING DIFFERENT EVALUATIONS 82 Suppose we have denned Rm-i- Consider pm-i,i, • • • ,Pm-i,m-i and 1 then tnere is a pm. If Pm-LiPm-iAT^i > m-i,s G An-i such that am_i)S / pm with common extension bm. Let ym = im_1)S.' Otherwise there is j G {1,... ,m — 1} such that pm-ij /. pm with common extension which we again denote bm. In this case let ym = xm-\. By symmetry 3 3 of X there are extensions p mi < p m_x xiox 1 <£ : n an and p^ m < b^ (Y? 1 < J < ) d a finite logarithmic measure xm 3 which is stronger than {xm-i,ym} such that for all j, £ p me lh xm < X. Then for every £ = 1,..., m let pm/ = (p^>€,... ,p^e) and let R'm = {(Pm/i^m)}™!- Just as in the base case let Am = {amtS : s € w} be a < maximal antichain in Cn — Ct({Pm,e}'eLi) such that for all s G u> s (1) 3i G u such that am>s < am_1;jS (2) 3xmtS stronger than xm-\^s such that for all j = l,...,n, Let R"m = {(am]S,xm,s)}sEa, and let Rm = R'mU R'^. With this the inductive construction id complete and we can define X — Um&UJRm- Let G be Cn-generic. Then GC\A\ contains some ai,s (or pltl) and so X[G](1) = xitS (resp. X[G](1) = x{). However for every j = 1,..., n, j J a{ s G G and so since a{s lh iiiS < X we have xljS < X[G ] (similarly Xi < X[GJ]). The same argument holds for every m G to. Indeed if J am,s G G n >lm, then X[G](m) = xm,s. But for all j = l,...,n, a ms € 0 3 J G and since a ms lh (xm?s < X) we obtain fTO)S = X[G](m) < X[G ']. Therefore -X"[G] is a pure condition which is a common extension of X[G^],...,X[Gn}. 4.5. EXTENDING DIFFERENT EVALUATIONS 83 The name X is Cn-symmetric. Consider any finite number of con 1 n ditions p ,... ,p and M G u. In the fixed enumeration of Cn for every j = 1,..., n there is ij such that pi = pijm Then there is k G u such that k > ij for all j and k > M. Then pktil < pi,... ,Pk,in < Pin and for Xk G Lk C LM for all j we have p*^. Ih "d^ G X". That is given any finite number of conditions p1,... ,pn and M £ LU there are exten 1 n sions qi < p ,... ,qn < p and a finite logarithmic measure x € LM such that V7 : 1 < Z < n, qi Ih x < X. Therefore the name X is C„-symmetric. D LEMMA 4.5.2. Le£ X — (X(i) : i G u) be a Cohen symmetric name for a pure condition, A an infinite subset of u and GQ a Cn- generic filter. Then there is a C„-symmetric name for a pure condition X* = (X*(i) :ieu) such that c (1) int(X*[G0}) QA or int(X*[G0}) C A and J (2) \Jmeu, j PROOF. Let {pm}me C logarithmic measure a;1)S such that int(a;i)S) C A or int(a;iiS) C ^4 and for all j = 1,..., n, a^ s Ih x\tS < Xi where ai)S = (a\ s,..., a™s). Let Ri = {(pi,i,xi)} U {{aits,xit3) : s G u}. Suppose we have defined 1 Rm-i. Consider pm-i,i, • • • ,Pm-i,m-i and pm. If pm±{pm_iiJ^ then 4.5. EXTENDING DIFFERENT EVALUATIONS 84 there is s 6 u such that pm JL am_i>s. In this case let bm be their common extension and let ym = xm-\,s- If there is j < m — 1 such that pm / Pm-i,j let &m be their common extension and let ym = xTO_i. Then there are extensions pm,e < pm-i,< for every £ = 1,... ,m — 1 and pm,TO < 6TO, and a finite logarithmic measure xm stronger than c {£m_i, ym} such that xm C A or xm C A and for all £ = 1,..., m, for 3 all j = 1,... ,n, p ml Ih xm < Xm where pTO,^ = (pj^,... ,p^e). Let ^4m = {fltii,s : s £ w} be a maximal antichian in C„ — C^lp™^}^) such that for all s € us (1) 3r* e w s.t. am>s < am_i,i<>, c (2) 3xm?s stronger than xm_1]is such that xmjS C ^4 or xmtS C A and for all j = l,...,n, a^s Ih xm)5 < Xm where om>s = lam,s> • • • i am,sJ u a Let i?m = {(Vm,t,xm))T=\ {( m,s,^m,s) : s e u}. With this the inductive construction is complete and we can define X = UmeuiRm. To see that X is symmetric consider any finite number of conditions p1,... ,pm in C„ and some M £ u>. Then \fj < m, 3ij G u> such that 3 p = pi. (in the fixed enumeration of Cn). There is k > M s.t. k > ij for all j < m. Then pkth < ph,... ,pk,im < pim and xk E Lk C LM are such that pk# \\- xk < X for all £ € {1,..., k} and so in particular Pk,e \y~xk 3 X[G ]. Then Gndp^ijUAt) contains some condition altS (or contains Pi,i). Then for every j < n, a{s Ih (fi)S < X) (resp. p{ ^ Ih (ii < X)) j and since a(s e G (resp. p^ € <3») xhs = X[G}(1) < X[&\ for 4.5. EXTENDING DIFFERENT EVALUATIONS 85 every j < n (resp. X\ = X[G](1) < X[G^\). The same argument holds for every Ak U {pk,i, • • • ,Pk,k}) and so X[G] is a common extension 1 n of XfG ],... ,X[G ]. Furthermore Xm = Uk>mRk is symmetric and for every Cn-generic filter in V[G] for every j < n we have XTO[G] < j Xm[G ]. Observe that Vi G u cn ll-cn "int(X(?)) C A or int(X(z)) C A . Let fl", f be C„-names such that lh "H = (X(i) : mt(X(i)) C i)" and lh "f = {X(i) : int(l(i)) C Ac)". Then following the proof of Lemma 4.4.6 obtain that for every p € Cn there is q < p such that H or T" is symmetric below q. Let E be a maximal antichian of conditions with this property. Then for every e £ E let X* \ e = H \ e if H is symmetric below e and let X* |" e = T f e if T is symmetric below e. For every e' G -EMe} let iee> : C+(e) = C+(e') be a partial order isomorphism. Then for every e' G E\{e} let X* \ e' = iet(,'(X* \ e). Then for every e & E, X* is a symmetric name for a pure condition such that 1 lh "int(X*) C A or int(X*) C Ac". Furthermore e lh X* < X and so in particular for all m E co, e lh X*m < Xm, where lh X*em = (X*(i) :i>m). Then X* = X*o where {e0} = G0 n £ is the desired symmetric name for a pure condition. • REMARK 4.5.3. Note that in V[G] the centered family extends the centered family Cj = {Xm[G^]}m&ul for every j = 1,..., n. CHAPTER 5 Preserving small unbounded families 5.1. Preprocessed Names for Pure Conditions M DEFINITION 5.1.1. Let T G [LU2] and let C be a centered family of (C(r)-symmetric names for pure conditions. We say that / is a good C(T)*Q(C)-name for a real if for every subset V o£u>2 such that rcF and centered family C of C(r')-symmetric names for pure conditions extending C, / is a C(i~") * <3(C")-name for a real. DEFINITION 5.1.2. Let C be a countable centered family of C(r)- names for pure conditions, let / be a good C(T)*Q(C)-name for a real, i:k G u, p G C(r), X a symmetric name for a pure condition in Q(C) such that lh k < minint(X). The name X is preprocessed for f(i), k, p and C where i, k € a> if for every v C k the following holds: If there is a countable centered family C of C(r)-symmetric names extending C, a symmetric name for a pure condition Y in Q(C) ex tending X and a condition A G A(/) such that (p, (v,Y)) < A then there is B G A(/) such that (p, (v,X)) < B. LEMMA 5.1.3. Let C be a countable centered family of C(T)-symmetric names for pure conditions, f a good C(r) * Q{C)-name for a real, X a symmetric name for a pure condition in Q(C). Let C be a countable centered family of symmetric names for pure conditions extending C 86 5.1. PREPROCESSED NAMES FOR PURE CONDITIONS 87 and Y a symmetric name for a pure condition in Q(C) extending X. If X is •pre-processed for f{i), k, p and C then Y is preprocessed for f{i), k, p and C PROOF. Let C" be a countable centered family of symmetric names for pure conditions extending C and let Z be a symmetric name for an extension of Y such that for some A E -4»(/) (p, (v, Z)) < A. However C" extends C, Z extends X, X is preprocessed for f(i), k, p and C, and so there is B E A(f) such that (p, (v,X)) < B. But Ih Y < X and so (p, (v,Y)) < (p, (v,X)) < B. Therefore Y is preprocessed for f(i), k, p and C. D LEMMA 5.1.4. Let C be a countable centered family of C(T)-symmetric names for pure conditions, f a good C(T) * Q(C)-name for a real, i, k E UJ, p e C(r), X a symmetric name for a pure condition in Q(C). Then there is a countable centered family of symmetric names for pure conditions C extending C and a symmetric name for a pure condition T" extending X, T' 6 Q(C) such that T" is preprocessed for f{i), k, p and C. PROOF. Let vi,... ,vs enumerate the subsets of k. The name Y and the centered family C will be obtained at finitely many steps. Consider (pi,(vi,X)). If there is a countable centered family C[ of symmetric names for pure conditions extending C and a symmetric name for a pure condition T[ E QiC^) extending X\k such that for some Ai E A{f), (p, (vi,T{)) < Ax let Ti = T{, d = C[. Otherwise let Ti = X, Ci = C. At step (s — 1) consider (p, (i»a,Ta_i)) and Cs-\. 5.1. PREPROCESSED NAMES FOR PURE CONDITIONS 88 If there is a centered family of symmetric names for pure conditions C's extending Cs_i, \C'S\ = |Cs_i| such that for some pure condition T's G Q(C'S) extending Ts_i, there is As G A(/) such that (p, (vs,T^)) < As let Ts = Va, Cs = C's. Otherwise let Ts = TB_i, Cs = C„-i. It will be shown that T" = Ts is preprocessed for f(i), k, p and C = Cs. Let v C k, C" a countable centered family of C(r)-symmetric names for pure conditions extending C", T" a symmetric name for a pure condition in Q(C") extending T' such that for some A G A(/)> (p,(u,T"\fc)) < A. Then w = Vj for some j G s + 1. Since C" extends C, C" extends Cj-\ and furthermore T" is a name for an extension of Tj_i. Therefore at stage j we have chosen a centered family Cj and a symmetric name for a pure condition Tj G Q(Cj) such that (p,(vj,Tj)) < Ai G A(/)- However lh T' < T,- and so (p,(v3,T')) COROLLARY 5.1.5. Let X be a C(r)-symmetric name for a pure condition in Q(C), where C is a countable centered family of C(F)- symmetric names for pure conditions and let f be a good C(T) * Q{C)- name for a real. Let {pj}jee be a finite number of conditions in C(r); k, n G LU. Then there is a countable centered family C of C(r)- symmetric names extending C, a symmetric name Y for a pure exten sion of X in Q(C) such that for all j < £ and i < n, Y is preprocessed for f{i), k, pj and C. PROOF. By Lemma 5.1.4 there is a countable centered family Co of C(r)-symmetric names for pure conditions, a C(r)-symmetric name XQ for an extension of X in Q(CQ) which is preprocessed for /(0), k, po 5.1. PREPROCESSED NAMES FOR PURE CONDITIONS 89 and C\. Again by Lemma 5.1.4 there is a countable centered family C\ of (C(r)-symmetric names extending Co and a symmetric name for a pure condition X\ extending X0, lh X\ G Q(C\) which is preprocessed for /(0), k, pi and C\. By Lemma 5.1.3 X\ is also preprocessed for /(0), k, po and C\. Repeating the argument £-times obtain a centered family of symmetric names for pure conditions Ce-i and a symmetric name X^_i G Q(Cg-i) such that for all j G £, Xi_x is preprocessed for /(0), k, pj and Ci-\. Repeating the argument above successively for /(l),..., f(n — 1) obtain a centered family C of symmetric names for pure conditions and a symmetric name for a pure condition Y G Q(C) extending X such that for all j E £, i € n, Y is preprocessed for f(i), k, pj and C. D COROLLARY 5.1.6. Let {pn}neuj enumerate C(T). LetC be a count able centered family of C(T) -symmetric names for pure conditions, X a C(r)-symmetric name for a pure condition in Q(C) and let f be a good C(r) * Q(C)-name for a real. Then there is a countable centered family C of C(r)-symmetric names for pure conditions extending C and a sequence (Yn : n G u) of OY)-symmetric names for pure conditions in Q(C) such that (1) II- Y0 < X and Vn G u lh Yn+1 < Yn (2) Vn G to\/i,j < n Yn is preprocessed for f(i), n, pj and C. PROOF. By Corollary 5.1.5 there is a C(r)-symmetric name for a pure condition Y0 extending X, such that Y0 G Q(C0) where C0 is a countable centered family of C(T)-symmetric names for pure con ditions extending C, which is preprocessed for /(0), po, 0 and Co- 5.1. PREPROCESSED NAMES FOR PURE CONDITIONS 90 Suppose Yn, Cn have been denned. Then by Corollary 5.1.5 there is a countable centered family Cn+i of C(r)-symmetric names for pure conditions extending Cn and a symmetric name for a pure condition Yn+i extending Yn such that for alii, j < n + 1, Yn+i is preprocessed for f(i), n, pj and Cn+i. Then C = Un€uiCn is a countable centered family of C(r)-symmetric names for pure conditions extending C, such that (Yn : n G LO) is contained in Q(C) and Vn G u, Vi,j < n, Fn is preprocessed for f(i), n, pj, and C". • COROLLARY 5.1.7. Let C be a countable centered family ofC(T)- symmetric names for pure conditions, f a good C(F) *Q(C)-name for a real and X a C(r) -symmetric name for a pure condition in Q(C). Then there is a countable centered family C of C(F)-symmetric names for pure conditions extending C and a C(r)-symmetric name Z = {Z{i) : i € to) for a pure condition in Q(C) such that Vn e u, Vi,j < n Zn = (Z(i) : i > n) is preprocessed for f(i), pj, n and C, where {Pn}neu is a fixed enumeration ofC(T). PROOF. Let C be a countable centered family extending C, (Yn : n G u>) a sequence of C(r)-symmetric names contained in Q(C) sat isfying Corollary 5.1.6. Passing to a subfamily we can assume that C = {X„}n€u where for all n G u, lh Xn+1 < Xn. Then using the fixed enumeration of C(r) obtain a (C(r)-symmetric name for a pure condi tion Z = (Z(i) : i G ui) such that for all n G us, II- Zn < Xn A Zn 5.2. Induced Logarithmic Measure LEMMA 5.2.1. Let {pi}^^ be a fixed enumeration of C(T) and let X = (X(i) : i G UJ) be a C(r)-symmetric name for a pure condition such that Vm G UJ, to,j < m the name Xm = (X(i) : i > m) is preprocessed for f{i), m, pj and C — {Xm}mebJ, where f is a good C(T)*Q(C) -name for a real. Then for every i,k G UJ and finite number of conditions p1,... ,pn in C(T) the logarithmic measure induced by the 7 set Vk(f(i), X, {p }™^) which consists of all x G [UJ] (1) pj Ih (x C int(X)) A (3j G UJ(X H int{X{j)) G X(j)+)), J (2) to c k3w v c x3AvJ G Ai(f) s.t. (Pj, (vUwi,X*))< AvJ where X* is a symmetric name for some final segment of X, takes arbitrarily high values. The conditions {Pj}"=i are said to witness the fact that x is positive. n PROOF. Let G be C* = Y\ i=l C(r;)-generic filter where to # j, 1 n Ti n Tj = 0 and Tt ^ r,, such that (p ,... ,p ) G G. Let w = A0 U • • • U AM-I be a partition of u into finitely many sets. By Lemma 4.5.2 there is a C*-symmetric name X = (X(i) : i G u>) such that for some jo G M mt(X[G]) C Ajo and for all m G w, j = l,...,n, Xm[G] = J (X(i)[G] : i > m) < Xm[G ]. Then in particular for all j = 1,... ,n, 3 Cj = {Xm[G ]}meu C (5(C) where C = {Xm[G]}mGa;. Since / is a good name, / is also a C* * Q(C)-name for a real. Let VI,...,VL enumerate the subsets of k. Fix j € {1,..., n} and s G {1,..., L}. Since fj = J 3 f/G is <5(C)-name for a real, there is qjS G G \ a C(r)-syrnmetric name 5.2. INDUCED LOGARITHMIC MEASURE 92 for a pure condition RjS in Q(C) and a finite subset UjS of to, such that J AjS = (QJS, (uJS, Rjs)) G A(/) and in V[G] the conditions (itjS, i?j5[G ]) and (vs,X[G}) are compatible with common extension (vs L)WjS,T[G]) (from Q{C)). Then in particular u>JS C int(X[G]) and vs UWJS\UJS C J int(i?jS[G ']). Since ^JS and X are names in Q(C), there is a C(r)- symmetric name for a pure condition ZjS (in fact a name for a final subsequence of X) in Q(C) which is their common extension. Then j there is tjS G G extending qjS and p> such that (tjs, (vs U Wja, ZjS)) < Ajs and (tjS, (VSUWJS, Zjs)) < (tja, (vsl)WjS, X)). In finitely many steps find a finite subset x of int(X[G]) such that for every s = 1,...,L and every j = 1,..., n there is WjS C x, a C(r)-symmetric name for a pure J condition ZjS in Q(C) such that lh ZjS < X, a Cohen condition tjS G G and a condition AjS G Ai(f) such that (tjS, (vs U WjS, ZjS)) < AjS and such that for some I E UJ, x f) iat(X(l)[G]) is X(/)-positive. Since X[G] < X[Gj] (for all j = l,...,n) we have that x C int(X[^']) and furthermore for every j = 1,..., n there is £,- G a; such that x D int(X(£j)[Gj]) is a positive subset of X(tj)[Gj). Then for every j — J l 1,..., n there is a condition pj G G extending p> and {tjS} s=i which forces "x C int(X)" and "xnint(X(^)) is a positive subset of Xtfj)". Since pj < tjs, then also we have that (pj, (vs U WjS, ZjS)) < AjS for all 5=1,...,£. Let JV be an integer greater than the indexes of pj for j = 1,..., n in the fixed enumeration of C(r) and also greater than i and max x. Let X* = Xjv- Recall that Z,-« is a C(r)-symmetric name for a pure con dition extending X and ZjS G Q(C). Then there is a (C(r)-symmetric 5.3. GOOD NAMES FOR PURE CONDITIONS 93 name for a pure condition Z*s in Q(C) such that lh "Z*s < Zjs and Z*s < X*" (in fact Z*s is a name for some final subsequence of X). However X* is preprocessed for f(i), max a;, {pj}j J there is a condition BjS G Ai(f) such that (pj, (vsUw s,X*)) < Bjs and so £ is a positive set. It remains to observe that x C Aj0 and so by the sufficient condition for arbitrarily high values (Lemma 2.1.10) the logarithmic measure induced by Vk(f(i),X,{p'}'j=1) takes arbitrarily high values. • 5.3. Good Names for Pure Conditions COROLLARY 5.3.1. Let{pi}ieuJ enumerate C(T) and let X = (X(i) : i G u>) be a C(T)-symmetric name for a pure condition such that Vm € wVi,j < m the name Xm = (X(i) : i > m) is preprocessed for f(i), m, Pj and C = {Xm}mew where f is a good C(r) * Q(C)-name for a real. Then there is a C(T)-symmetric name for a pure condition Y = (Y(i) : i € u) such that (1) Vm £ u, Ym = (Y(i) : i > m) extends Xm, and (2) for all i G u, v C i, p G C(r), s G [u] + s G Y(i) there is wv C s and A G Ai(f) such that (p, (v U wv,Yi+1)) < A. PROOF. For every p G C(r) let C(p) = C(r)(p). By Lemma 5.2.1 there is Xi G V\{X\, f(l),pi) with witness pi^. Fix a maximal an- tichain A\ = {ai)S : s G u>} in P — C(jp\j) such that for every s G UJ, 5.4. UNBOUNDEDNESS 94 aljS witnesses that x1)S is in V1(X1, f(l),als). Let Ri = {(Pn, £i)} U {(als, xls:s£ ui}. Suppose Rm-i is defined. If Pm-L{Pm-\,i)i Pm-ij with common extension which we again denote bn. In this case let ym = xm-\. Then by Lemma 5.2.1 there is a measure xm 1 in Vm(Xm, f(m), {Pm-ij}^ U {bm}) which is stronger than xm_x and yTO. Thus in particular there are extensions pmj, < pm-i,i iori < m — 1 and pm,m s (1) 3i G w(am)S < am_i,^) (2) there is a finite logarithmic measure xmtS stronger than xm-1:ia such that am:S witness that xm^s is in Vm(Xm, f(m), ams). Let Rm = {(PmhXrr,)}^ U {{ams,xms) : s G u} and let Y = UmeaJi?TO. Then Vm G u>, Ym = (Y(i) : i > m) extends Xm and has the desired properties. • 5.4. Unboundedness THEOREM 5.4.1. Let C be a countable centered family of C(r)- symmetric names for pure conditions, let F be a countable subset ofu>2, let f be a good C(T)*Q(C)-name for a real and S G k>i\I\ Let h = UG$ where Gs is the canonical name for the C({6})-generic filter. Then 5.4. UNBOUNDEDNESS 95 there is a countable centered family C of C(T U {5})-symmetric names for pure conditions which extends C and such that for every centered family C" of C(t02)-symmetric names for pure conditions which extends C, H-c(w2)*Q(C") "h •£* /". PROOF. We can assume that C = {Ym}m€u}, where Ym = (Y(i) : i > m) and Y = (Y(i) : i £ LJ) is the C(r)-symmetric name constructed in Corollary 5.3.1. Let g be a C(r)-name for a function in "u such that Vp 6 C(r)Vz € u, p Ih g(i) = k if and only if k = max{j :vCi,wE H (p, {y U w, Y)) < A for some A € A%(f) and A Ih «/(*) = ]"}. Let J be a C(r U {£})-name for a subset of ui such that U-J = {i : g(i) < h(i)} and for every m £ u let Zm be a C(r U {<5})-name such that Ih Zm = (Y(i) :i>m and i € J). CLAIM. For all m £ a; the name Zm is C(F U {5})-symmetric. PROOF. Let po,.. • ,pn-i be a finite number of conditions in C(r U {5}) and let M € w be given. Then for every i £ n pi — p® Up] where v\ ~V% [Txw and p\ =Pi \ {6} x u. By construction of Y there are extensions q® < p° and a finite logarithmic measure x £ LM such that 5.4. UNBOUNDEDNESS 96 \fi £ n, q? lh x = Y{£) where £ > m, £ > M and £ > max{j : (S,j) £ domain(p*), i £ n}. Furthermore for every i £ n there is t° G C(r) extending q® such that *i* "~ 5 CO ~ fa for some fcjGw. Then let L > maxien fcj and for every i € n let tl=p]u{((8,£),L)}. Then ^ = if U t\ < p{ and U lh "Y" (€) =xAf>mA£eJ". That is tj lh x < ZTO. Therefore Zm is symmetric. D Then let C = {Zm}meM and let Z = Z0. Consider arbitrary cen tered family C" of C (^-symmetric names such that lh C C Q(C"). It is sufficient to show that Va £ [w] nn lhc(w2) «(a,Z) lhe(c„) «3i > *(/(i) < h(i) since ll"C(<«) "{(«, Z):a£ [LV\ Let a € M p lh "£ £ J and s = int(i?) n int(Z(£)) is Z(£)- positive". 5.4. UNBOUNDEDNESS 97 By definition of Z{t) there is w C s and A e At(f) such that (p,(bUw,Y)) (p,(bUw,R))H- uf(e) < g(t) < H*)"- D DEFINITION 5.4.2. Let P be the partial order of all pairs p = (Tp, Cp) where T is a countable subset of U2, Cp is a countable centered family of C(rp)-symmetric names for pure conditions with extension relation defined as follows: p < q if Tq C Fp and lt~c(rp) Cq C Q(CP). The partial order P is countably closed and adds a centered family of C(o;2)-symmetric names for pure conditions CH = U{CP :peH} where H is P-generic. By Lemma 4.4.6, forcing with Q(CH) over yPxC(W2) adds a real not gplit by c(W2) y n [u]" = BCMXP n [uf. To see that the first tui Cohen reals remain an unbounded family con sider an arbitrary C(o;2) * (3(CH)-name / for a real. Then there is a condition p G H such that / is a C(rp) * Q(Cp)-name for a real. Then for every q < p either there is a further extension a such that / is not 5.4. UNBOUNDEDNESS 98 a C(ro) * Q(Co)-name for a real, or / is a good C(ro) * <3(Ca)-name. Then let A — A~ U A+ be an antichian of conditions which is maximal below p and such that Va € A~ f is not a C(ra) * Q(Ca)-name for a + real and Va € A f is a good C(ra) * (3(Ca)-name. Since p E H and / is a C^) * (2(C#)-name for a real, there is a E H n A+. That is there is a e if such that / is a good C(r„) * Q(Ca)-name for a real. Let 7i be the collection of all names h such that h = UG$ where 5 E u)x and G^ is the canonical C({5})-name for the C({5})-generic filter (that is 7i is the set of the first oj\ Cohen reals). Then by Theorem 5.4.1 the set Dj = {qeF:3he H(q lb " ^c(u»)*Q(c6) "h £* /"")} where H is the canonical P-name for the P-generic filer, is dense below a. Therefore there is h E Ji such that LEMMA 5.4.3. Let Ti,^ be countable subsets of U2 such that Fj D 1?2 = 0 and let i : F\ = F2 be an isomorphism. Let X be C(ri)- symmetric name for a pure condition. Then there is C(rx U T2)- symmetric name for a pure condition X such that lhC(riUr2) "X < X and X < i(X)". IfY and Z are OF 1)-symmetric names for pure conditions such that 11- "X < Y and X < Z" then U lb(riur2) X We will need the following claim. 5.4. UNBOUNDEDNESS 99 CLAIM. Let p\,... ,pk be any finite number of conditions in C(ri U T2) and let M Gw. Then there is a finite logarithmic measure x G LM and extensions qi < pj,..., q^ < pk such that Vi < k, ftlh ux PROOF. Note that \fi < k,pi = p\Up? where p\ = Pi \ Tj, pf = pt \ _1 F2. Let q] = p\ and q? — i (p?). Then since X is C(r1)-symmetric there is x G LM and extensions q}Y < q\ and qf2 < gf such that Vi < fc, g£x Ih x < X and g^ II- x < X. But then i(gfa) lhC(ra) x < i(X) and so r» = q}^ U gfj is an extension of pi such that r; lhc(riur2) "^ < iandi With this we can proceed with the proof of Lemma 5.4.3. PROOF. Fix an enumeration {pn}n6w of C(ri ur2) and inductively construct a C(Ti U r2)-symmetric name X such that for all n G w, l^c(riur2) X{n) Suppose Y and Z are C(ri)-symmetric names for pure conditions such that lhc(ri) X < Y AX < Z. Then H-C(r2) i(X) < i(Z) and so ^c{TlUv2)X Begin with a model of CH and consider a subset {pi : i € /} of P of size K2. By the Delta System Lemma there is a subset J oi I, \J\ = H2 such that {Ti : i £ J} forms a delta system with root A where \/i € I(Ti — TPi). Furthermore J might be chosen so that for all i,j E J there is an isomorphism a^ : Pi(l) = pj(l), such that a*,- f A 5.4. UNBOUNDEDNESS 100 is the identity and such that Cj = CPj = {atij(X) : X € CPi}. If A = 0 then by Lemma 5.4.3 for every X E Ci where Cj = CPi, there is C(rj U rj)-symmetric name for a pure condition Xx extending X and Oij(X) and so pk = (Tk, Ck) where T^ = Tj U Fj and is a common extension of pi and pj. However as mentioned in section 4.1 if the root of the delta system is non-empty the above argument does not hold and a stronger combinatorial property on the names for pure conditions is needed. CHAPTER 6 A look ahead 6.1. General Definition of Symmetric Names DEFINITION 6.1.1. Let X be a C(r)-name for a subset of UJ, where r 6 [UJ-^ . Then X is symmetric if for all finite subsets V = {70,..., jn} of u>2, where 70 = minT < 71 < • • • < 7« = supT, for all finite fami lies of conditions (p$ )*<*,,• Q C(T fl TJ-YXJ-I) for j = 1,..., n and every M E u, there is m > M which belongs to n 2 ,rn71\70( i2 =ihullp2^rn72^7i(... ni"=1hullpni!rn7n\7n_1 (X)...)). REMARK 6.1.2. Note that X is a C(r)-symmetric name for a subset of UJ if and only if for all finite subsets V = {70,..., 7„} of u2, where 70 = min r < 71 < • • • < jn = sup T, for all finite families of conditions (p* )»<*:• C C(r fl 7?\7j-i) f°r i = 1,..., n and every M E UJ, there is a tree of extensions $ = {^(it. ..ij) : I < j 101 6.1. GENERAL DEFINITION OF SYMMETRIC NAMES 102 matrix of conditions and to the tree <5 = 3>(P) as an associated tree of extensions. Note that definition 4.2.2 coincides with the particular case of the above definition in which T is a singleton. This definition generalizes to names for pure conditions. DEFINITION 6.1.3. Let X be a C(r)-name for a pure condition, where T is a countable subset of ujf Then X is symmetric if for all finite subsets V = {70,..., jn} of 1V2, where 70 = minT < 71 < • • • < 7„ = supT, for all finite families of conditions (p'i)i rt=1hullp>1 . ,rn7i\7o (rt=1hullp? ,rn72\7i (...nf;=1hullp? (X)...)). REMARK 6.1.4. Note that X is a C(r)-symmetric name for a pure condition if and only if for all finite subsets r" = {70,... ,7„} of 0J2, where 70 = minT < 71 < • • • < jn = supT, > for all finite families of conditions {p i)i P = (pDij as a matrix of conditions and to the tree $ = $(P) as an associated tree of extensions. Note that definition 4.3.2 coincides with the particular case of Definition 6.1.3 in which T is a singleton. 6.2. The H2-chain condition Having in mind the construction following Definitions 5.4.2 consider the following Lemma: LEMMA 6.2.1. Let T and 6 be countable subsets ofoj^, let A = Tn©; S] = TUB and let X be C(T)-symmetric name for a pure condition. Suppose sup A < min T\ A < sup T\ A < min 0\ A and let i : V = G be an isomorphism such that i \ A = id. Then for every finite subset V = {70,. • •, 7S} of u>2 where 70 = min 0 < 7J < • • • < 7S = sup Q, and all finite families of conditions (p^)*<*;. Q C(fi D Jj\jj-i) for j = 1,..., s and every M € u there is x E LM and an associated tree of extensions $(P) = {^(zi ...ij):\ 4>\\- ux Here P denotes the collection {pDij of the given Cohen conditions. 6.2. THE N2-CHAIN CONDITION 104 PROOF. Adding more ordinals if necessary we can assume that r n e A V n A = {7,-Wl, V n T\A = {jj}je{n,2nb ' \ = Mj^nMl Furthermore we can assume that i(jj) = jj+n for all j 6 (n, 2n] and 7„ = sup A, 72„ = supT\A, 73n = sup 6. Also for all j G (0,3n] let kj = A;. Let AfGwbe given. We have to obtain a tree of extensions $(P) associated with the matrix P and a finite logarithmic measure x G LM such that the maximal nodes of $ force that x extends X and the isomorphic copy i(X). For every j G (0,2n], i € (0, k] let r? = p\ and for j G (n, 2n], z € (k, 2k] let r^ — «-1(Pj*£). Then i? = (r^)jj is a matrix of conditions in C(r) and so by symmetry of X there is a finite logarithmic measure x G LM and a tree of extensions \P(-R) = {ip(ii • • • ij) : 1 < j < 2n, 1 < ij < fcj-} where for j G (0, n] kj = A; and for j G (ra, 2n] A^- = 2A;, the maximal nodes of which force that x extends X. For j G (0,2n] let 4>(ii---ij) =ip(ii...ij) and for j G (2n, 3n], say j = In + m let 0(ii. ..ij) = z('0(?1...i„;zn+i + k,... ,in+m +k)). Then let $(P) = {0(i!... ij) : 1 < J < 3n, 1 < ij < A;} 6.2. THE K2-CHAIN CONDITION 105 where 4>(ii) = Then 4> — (a0,ax,a2) where a0 = 4>{i1...in) = ${ix...in), aa = (4>(ii ...ij):n 1 (aQ, i~ (a2)) = 4>(ij -..in; kn+\ + k,... ,i3n + k) are maximal nodes of ^f(R) and so they force that x extends X. It remains to observe that i(ao,i~1(a2)) = (ao,a2) since i \ A = id and a so (cuo, a2) \\- x < i(X). Therefore 4> Ih x < X and x < i(X)". D LEMMA 6.2.2. Let F andQ be countable subsets ofu2, let A = m6; n = rue, sup A < min T\A < sup T\A < min 6\A and let i : T = 0 be an isomorphism such that i \ A = id. Let X be C(r)-symmetric name for a pure condition. Then i(X) is a C(0)- symmetric name for a pure condition and there is a C(Q)-symmetric name for a pure condition X such that \\-c{n)X PROOF. Enumerate all finite subsets of u2 and associated matrices of conditions on Q, such that each pair is enumerated cofinally often. At 6.3. CONCLUSION AND OPEN QUESTIONS 106 stage n consider the n-th pair and let mn be an integer grater than the measures and domains of all finite logarithmic measures that have been defined up to this stage. Apply Lemma 6.2.1 to this n-th pair and the integer mn to obtain a corresponding tree of extensions Tn and a finite logarithmic measure x E Lmn such that the maximal nodes of the tree u force x < XAx < i{X). Then let {(t, x) : t max node of Tn} CI D 6.3. Conclusion and open questions A family A of infinite subsets of u, with pairwise finite intersection is an almost disjoint family. An almost disjoint family which is maxi mal, is called a maximal almost disjoint family, usually abbreviated as mad family. The almost disjointness number a is the minimal size of a maximal almost disjoint family. The ultrafilter number u is the mini mal size of an ultrafilter base. A family T of subsets of uo has the strong finite intersection property if the intersection of any finite subfamily of T is infinite. A pseudo-intersection of a family T is an infinite set al most contained in every element of the family. The pseudo-intersection number p is the minimal size of a family which has the strong finite intersection property and no pseudo-intersection. THEOREM 6.3.1 (GCH). Let K be a regular uncountable cardinal. Then there is a ccc generic extension, in which b = K PROOF. In [12] J. Brendle shows that if V is a model of ZFC*, K is a regular uncountable cardinal, (/Q : a < K) is a <*-well ordered sequence of strictly increasing functions from wtow and in V, c = K, K + 2 = K and (fa : a < K) is unbounded and A is a maximal almost 6.3. CONCLUSION AND OPEN QUESTIONS 107 disjoint family, then there is a ccc forcing notion F(A) of size c such that \\~r(A) ""4 is not mad and (fa : a < K) is unbounded". Using an appropriate bookkeeping device, along the finite support iteration of Theorem 3.6.3, one can destroy all mad families of size < K. • THEOREM 6.3.2 (GCH). Let K be a regular uncountable cardinal. Then there is a ccc generic extension in which p = b = K PROOF. Along the finite support iteration from the proof of Theo rem 3.6.3, one can force with all cr-centered forcing notions of size < K, and so provide that in the final generic extension MA THEOREM 6.3.3 (GCH). Let K be a regular uncountable cardinal. Then there is a ccc generic extension in which p = b = Ac PROOF. Modifying an argument from A.Blass and S. Shelah [9], it will be shown that in the model of Theorem 3.6.3 the ultrafilter number u = K+. Suppose u < K+ and let T be an ultrafilter base of size u. Then there is /? < K+ such that fCV^ = ^[^73] where as usual, for + every 7 < K G1 — G (1 P7. We can assume that in 1/g, Qa = Q(Ca) where a = f3 + 1 for an appropriate centered family of pure conditions Ca. Let sa - U{u : 3T(u,T) € G} where Ga = G0*G, i.e. G is Q(Ca)- generic over Vp = V[Gp\ and let X = {n : \sa n n\ is even}. Then X £ W V^Ga] H [OJ] - Let X and sa be Q(CQ)-names for X and sa respectively, 6.3. CONCLUSION AND OPEN QUESTIONS 108 in V[Gr/j]. It will be shown that neither X nor its complement contain infinite set from Vp, which contradicts the hypothesis that T is an ultrafilter base. Suppose to the contrary, that there is Y £ ^ fl [w]u c such that V[Ga] N Y C X or V[a] N Y C X . Without loss of generality suppose V[Ga] N Y C X. Then there is (u,T) € Q(Ca) such that (tt,T) lh Y" C X. Let m = minint(T) and let y e y such that y > m. Then (u,T\y) and (tiU{m},T\j/) extend (u,T). However (u, T\y) \\- saC\y = u and (u U {m}, T\y) \\- sa(ly = uU {m}. Then one of those extensions forces "y 0 X", which is a contradiction. • COROLLARY 6.3.4 (GCH). Let K be regular uncountable cardinal. Then there is a ccc generic extension, in which p = t=f) = b = K; and QUESTION 6.3.5. What can be said about x, t, g in this model? In section 4.2, we showed that in the Cohen extension, the collection of all subsets of OJ which do not have symmetric names forms an ideal Insym- This ideal has a very natural definition, and on the other hand its properties seem to be distinct from the properties of known ideals. QUESTION 6.3.6. Find a generating set for Insym- Is there an ab solute analogue of Insym? What are the properties ofV{u))/Insym? One can combine the techniques of chapter III with techniques of S. Shelah, from his original paper [31] on the consistency of b — ui\ < s = a = co2 to obtain a forcing notion which preserves a given unbounded family unbounded, which destroys a maximal almost disjoint family 6.3. CONCLUSION AND OPEN QUESTIONS 109 and adds a real not split by the ground model reals. Furthermore, it seems reasonable to expect that the construction of the countably closed, N2-c.c. forcing notion from chapter V, see Definition 5.4.2, can be modified to obtain a countably closed, K2-c.c. forcing notion (or al ternatively ^-closed, «;+-c.c.) which adds a centered family C of C(A)- names for pure conditions, such that Q(C) preserves all unbounded families unbounded, destroys VC(A) n [UJ]" as a splitting family, and adds a real almost disjoint from the elements of a given maximal al most disjoint family in VC(XK Then it becomes imperative to find an appropriate way to iterate this forcing notion and obtain the consis tency oib = K [I] U. Abraham Proper forcing, for the Handbook of Set-Theory. [2] B. Balcar, J. Pelant, P. Simon The space of ultrafilters ofN covered by nowhere dense sets, Fund. Math., vol. 110(1980), pp. 11-24. [3] T. Bartoszyriski and H. Judah Set theory: on the structure of the real line, A.K. Peters, 1995. [4] T. Bartoszynski and J. Ihoda [H. 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