{"@context":{"@language":"en","Affiliation":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","AggregatedSourceRepository":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","Campus":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","Creator":"http:\/\/purl.org\/dc\/terms\/creator","DateAvailable":"http:\/\/purl.org\/dc\/terms\/issued","DateIssued":"http:\/\/purl.org\/dc\/terms\/issued","Degree":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","DegreeGrantor":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","Description":"http:\/\/purl.org\/dc\/terms\/description","DigitalResourceOriginalRecord":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","FullText":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","Genre":"http:\/\/www.europeana.eu\/schemas\/edm\/hasType","IsShownAt":"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt","Language":"http:\/\/purl.org\/dc\/terms\/language","Program":"https:\/\/open.library.ubc.ca\/terms#degreeDiscipline","Provider":"http:\/\/www.europeana.eu\/schemas\/edm\/provider","Publisher":"http:\/\/purl.org\/dc\/terms\/publisher","Rights":"http:\/\/purl.org\/dc\/terms\/rights","ScholarlyLevel":"https:\/\/open.library.ubc.ca\/terms#scholarLevel","Title":"http:\/\/purl.org\/dc\/terms\/title","Type":"http:\/\/purl.org\/dc\/terms\/type","URI":"https:\/\/open.library.ubc.ca\/terms#identifierURI","SortDate":"http:\/\/purl.org\/dc\/terms\/date"},"Affiliation":[{"@value":"Science, Faculty of","@language":"en"},{"@value":"Mathematics, Department of","@language":"en"}],"AggregatedSourceRepository":[{"@value":"DSpace","@language":"en"}],"Campus":[{"@value":"UBCV","@language":"en"}],"Creator":[{"@value":"Sandberg Maitland, William","@language":"en"}],"DateAvailable":[{"@value":"2010-02-21T18:40:22Z","@language":"en"}],"DateIssued":[{"@value":"1977","@language":"en"}],"Degree":[{"@value":"Master of Science - MSc","@language":"en"}],"DegreeGrantor":[{"@value":"University of British Columbia","@language":"en"}],"Description":[{"@value":"We let:\r\nZF = the Zermelo-Fraenkel axioms of set theory without the Axiom of Choice\u201e(AC) . ZFC = ZF + AC .\r\nI = \" There exists an inaccessible cardinal \" .\r\n\u03c8 = \" Every set of reals definable from a count able sequence of ordinals is Lebesgue measurable \".\r\nDC = the Axiom of Dependent Choices.\r\nLM = \" Every set of reals is Lebesgue measurable\r\nIn 1970, Solovay published a proof by forcing of the following relative consistency result:\r\nTheorem \r\nIf there exists a model M of ZFC + I, then\r\nthere exist extensions M [G] and N of M such that:\r\n(a) M [G] |= ZFC + \u03c8.\r\n(b) N I= ZF + DC + LM .\r\nBoolean-valued techniques are used here to retrace Solovay's proof on a different foundation and prove the following result:\r\nTheorem \r\nLet IK be a non-minimal standard transitive\r\nmodel of ZFC + I. Then:\r\n(a) IK |= there is a model of ZFC + \u03c8\r\n(b) IK |= there is a model of ZF + DC + LM .","@language":"en"}],"DigitalResourceOriginalRecord":[{"@value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/20657?expand=metadata","@language":"en"}],"FullText":[{"@value":"A BOOLEAN-VALUED APPROACH TO THE LEBESGUE MEASURE PROBLEM by WILLIAM \/SANDBERG MAITLAND B_.Sc, University of B r i t i s h Columbia, 1973 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n 4 THE FACULTY OF GRADUATE STUDIES (Department of Mathematics \u2022) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December, 1976 \u00a9 William Sandberg Maitland, 1976 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced deg ree a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t ha t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depar tment o f Mathematics The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date January 15 , 1977. i Abstract We l e t : ZF = the Zermelo-Fraenkel axioms of set theory without the Axiom of Choice\u201e(AC) . ZFC = ZF + AC . I = \" There exists an inaccessible cardinal \" . SV = \" Every set of reals definable from a count able sequence of ordinals i s Lebesgue meas-urable \". DC = the Axiom of Dependent Choices. LM = \" Every set of reals i s Lebesgue measurable In 1970, Solovay published a proof by forcing of the following r e l a t i v e consistency r e s u l t : Theorem If there exists a model M of ZFC + I, then there e x i s t extensions JMC [G] and DN of JMC such that: (a) JMC [G] |= ZFC + \u00a5 (b) 3N [= ZF + DC + LM . Boolean-valued techniques are used here to retrace Solovay's proof on a d i f f e r e n t foundation and prove the following r e s u l t : Theorem Let IK be a non-minimal standard t r a n s i t i v e model of ZFC + I. Then: I V (a) IK |= th e r e i s a model of ZFC + \u00a5 (b) IK |= th e r e i s a model of ZF + DC + LM . V Contents Abstract Acknowledgements Introduction 1 1 1 v i 1 Section 0 : Section 1 : Section 2 : Section 3 : Section 4 : Section 5 : Section 6 : Conclusion Bibliography Boolean-valued models and generic extensions 7 The Lebesgue Measure Problem and the Axiom of Choice 42 Some model theoretic properties of Lebesgue measure 4 7 The random reals 63 The Levy algebra 71 The model of Levy 91 The model of McAloon 103 114 116 v i Acknowledgements My sincere thanks go to Dr. Andrew Adler, not only for his suggestion of the topic of t h i s thesis and willingness to oversee the work, but e s p e c i a l l y for his encouragement and patience during the long course that has lead to i t s completion. His advice and insights have had a decisive influence on t h i s presentation of the topic. I wish also to acknowledge the support the Mathematics Department has given me. Special thanks are due to Mrs. Anne Rogers for her hard work and diligence i n typing the manuscript. I am grate f u l to Mr. Ralph Pudritz for his helpful com-ments on the manuscript, and to Mr. Peter Lomax and Ms. Miriam Silbermann for t h e i r help with tr a n s l a t i o n s . And l a s t l y , my acknowledgements go to the many who have shown f r i e n d l y concern and encouragements, not the least of whom i s Mr. Anatole Lebovich. 1 Introduction Under what hypothesis can we consistently assume that a l l sets of reals are Lebesgue measurable? This i s the essence of the Lebesgue measure problem. Here we present a recent set t h e o r e t i c a l investigation of t h i s problem due to Solovay. Lebesgue measure i s countably additive and t r a n s l a t i o n invariant. Under the hypothesis that the reals can be well-ordered, these two properties allowed early researchers ( e.g. V i t a l i , Bernstein ) to construct various sets of reals which are not Lebesgue measurable. Set functions with domain the powerset JP (3R ) of the r e a l s , which drop one of the above two constraints, have been the focus of some attention. But such would-be measures cannot compete with Lebesgue measure for i t s central role i n modern r e a l analysis. If we must accept the presence of non-measurable sets i n ordinary analysis, then i t would be useful to know how much t h e i r existence depends on the Axiom of Choice (AC). Solovay's research, published in 1970, shows that i t i s consistent with the Zermelo-Fraenkel axioms of set the-ory (ZF) and the P r i n c i p l e of Dependent Choices (DC) to assume that a l l sets of reals are Lebesgue measurable (LM), given the consistency of ZF+AC together with the statement (I) that a ( strongly ) inaccessible cardinal e x i s t s . This main theorem of Solovay indicates that i t i s impossible to prove the existence of non-measurable sets from the ZF 2 axioms with DC, provided that the theory ZF+AC+I i s con-si s t e n t . Hence the existence of non-measurable sets i s dependent on some form of AC which i s stronger than DC. To prove t h i s r e l a t i v e consistency r e s u l t , a model jN of ZF+DC+LM i s constructed from a model M of ZF+AC+I. Solovay's construction uses an unramified form of the forcing method, e s s e n t i a l l y due to Cohen. His main inn-ovation i s the use of Borel sets of po s i t i v e measure to replace Cohen's f i n i t e forcing conditions. Solovay has conjectured that the hypothesis regard-ing the consistency of I i s dispensable, though no proof has been forthcoming as yet. In 196 9, Sacks published an account of another Solovay r e s u l t , using ramified langua-ges and a measure theoretic forcing argument. He demon-strated that i f ZF i s consistent, then ZF+DC+ \"there ex-i s t s a countably additive, t r a n s l a t i o n invariant extension of Lebesgue measure on jP'(3R)\" i s consistent. The state-ment i n quotes i s weaker than LM, but i t s consistency re-quires no hypothesis regarding I. In 1965, Solovay and Scott, and independently Vopenka, noticed that forcing arguments could be translated into constructions involving so-called Boolean-valued models. Our presentation of Solovay's 1970 research begins with a resume of t h i s method of constructing a generic extension of a given ground model without forcing. A close examina-t i o n of [23] ( e.g. pp. 31 - 8, pp. 49 - 50 ) leaves no 3 doubt that Solovay's o r i g i n a l conception of his work on the Lebesgue measure problem was in terms of Boolean-valued models, rather than the c l a s s i c a l forcing arguments which predominate his f i n a l published account. The use of Boolean-valued methods provides a more natural and i n t u i -t i v e development of Solovay's ideas. In our i n i t i a l sec-t i o n , we have concentrated on those aspects of Boolean-valued models which have a d i r e c t bearing on Solovay's 1970 constructions, and have t r i e d to improve and complete some of the standard proofs i n t h i s area. In Section 3 we describe Solovay's notion of a random r e a l , which i s his main innovation mentioned above and the notion which motivated the Solovay-Scott development of Boolean-valued models. In t h i s , we have started with Solo-vay's d e f i n i t i o n , and translated his development into the language of Section 0. A d i f f e r e n t development ( based on D e f i n i t i o n 3.5 ) i n the same Boolean language can be found i n [9]. Lemma 3.8 establishes the equivalence of the two approaches. In t h i s exposition we have endeavored to combine the i n t u i t i v e c l a r i t y of the Boolean-valued approach with a rigorous foundational background. Two points are often glossed over i n presentations of t h i s type. It i s often not s p e c i f i e d whether the models involved are sets or classes. This lack of precision opens any treatment of definable sets to the p o s s i b i l i t y of set t h e o r e t i c a l para-4 doxes. To combat t h i s , we consider a l l our model construc-tions to take place within a model which i s a set, rather than within the \"real world\" of Solovay. The second s t i p -u l a t ion i s that our ground model i s countable. The reason-ing behind t h i s i s explained in Section 0. These two points ensure that our model constructions rest on an e x p l i c i t and correct foundation. The concluding sections describe the two main models of Solovay, whose o r i g i n he a t t r i b u t e s to Levy and McAloon. The l a t t e r model i s usually defined v i a the eight funda-mental Godel operations ( see [27] ), or by an extended form of the Reflection P r i n c i p l e ( see [15] ). Because of our adherence to models which are sets, we have been able to employ Godel-numbering to present a simpler construction needing much less background. The LeVy model i s construc-ted from an unpleasant Boolean algebra L which i s the sub-je c t of Section 4. We have f i l l e d i n some of the necces-sary technical work in t h i s LeVy algebra that i s avoided elsewhere. By a c l a s s i c a l algebraic argument, a theorem of Jensen ( [9], p. 76 ) i n d i r e c t l y implies that L i s homo-geneous. In our Lemmas 4.8 and 4.10 we have modified Jensen's theorem considerably, providing a d i r e c t proof of the homogeneity of L. Another often neglected point i s the problem of f i r s t -order d e f i n a b i l i t y of sets. In addressing t h i s topic, we have included relevant material here which i s usually only 5 alluded to. In Section 5, for example, we introduce the notion of uniform d e f i n a b i l i t y . This has other names in the l i t e r a t u r e ( e.g. \" s p e c i f i a b i l i t y \" , [4] ), but there seems to be no standard usage. In t h i s , we d i f f e r from Solovay's development which uses his M - J R - d e f i n a b i l i t y ( [23] p. 41 ). The somewhat informal use of d e f i n a b i l i t y i n Section 5 i s formalized i n Section 6. Here we show that the source of our d e f i n a b i l i t y problems i s Richard's paradox. While the McAloon model substantiates the main theorem of Solovay quoted above, the Levy model gives an equally in t e r e s t i n g secondary theorem of Solovay: If ZF+AC+I i s consistent, then i t i s consistent with ZF+AC to assume that every set of reals definable from a countable sequence of ordinals i s Lebesgue measurable. Using the formulas of set theory, we cannot e x p l i c i t l y define a non-measurable set without an uncountable sequence of ordinals. These notions are made precise i n Sections 5 and 6. Section 2 deals with some absoluteness properties of Lebesgue measure. For t h i s work we have selected a notion of absoluteness due to Shoenfield that i s naturally adap-ted to the model extension process. In most other respects, the development of t h i s section follows Solovay. We d i f f e r from the Solovay development by f i r s t establishing that the property of being a set of Lebesgue measure zero i s absolute. The main lemma of Solovay ( Lemma 1.6.4, p. 31, 6 [23] ) follows e a s i l y as our Corollary 2.25. Some prefatory remarks on the use of the Countable Axiom of Choice (AC^) i n analysis are included i n Section 1. Our aim here i s to emphasize that the r e a l impact of DC on ordinary analysis i s through AC^. Our set theoretic and model theoretic notation and nomenclature i s standard for the most part, being consis-tent with that of [1] and\/or [14]. 7 Section 0 : Boolean-valued models and generic extensions Our approach to the proof of Solovay i s based on the concept of a Boolean-valued model of set theory. We st a r t here with a f a i r l y general treatment of t h i s subject, then in l a t e r sections select the l i n e of application that leads to the Solovay r e s u l t . To begin, we r e c a l l some background. As usual, ZF i s used to stand for the Zermelo-Fraenkel set theory, i . e . the c o l l e c t i o n of theorems that follow from the Zermelo-Fraenkel axioms ( less the Axiom of Choice ). ZFC denotes the f u l l c o l l e c t i o n of theorems following from ZF and the Axiom of Choice (AC). For t h i s section the terms \"set theory\" and ZFC w i l l be used synonymously. A model of set theory i s an ordered pair M = (M,E), where M i s a set and E i s a binary r e l a t i o n on M ( i . e . E c M2 ) which s a t i s f i e s a l l the axioms of set theory as the interpretation for ' e 1 . The symbolism DM j= , can be defined by induction on the complexity of cf) ( see [1] ) . M i s referred to as the universe or underlying set of M. In s t i p u l a t i n g that M i s a set rather than a class, we w i l l avoid the danger of set theoretic paradoxes which might otherwise impair the model construction process. However, many popular versions of the theorems we w i l l use assume M to be a clas s , and care must be taken when we meet such theorems. 8 D e f i n i t i o n 0.1 (a) A binary r e l a t i o n R i s well-founded i n a set H <= dom(R) i f there i s no se-quence fx ) cr H such that x n R x holds ^ n n+\u00b1 n for each n e w . (b) A model M = (H,R) i s extensional i f for a l l x, y, and z i n H, ( zRx -H- zRy ) -> ( x =' y ) . (c) A model M = (H,R) i s standard i f R <= H 2 H e . (d) A model M = (H,R) i s t r a n s i t i v e i f H i s a t r a n s i t i v e set, i . e . for each x e H, x c: H . In 0.1 (c) above, we assume that there i s a \"real world\" of sets, and that e i s the natural membership re-l a t i o n . The statement that ZFC has a model implies the state-ment that ZFC i s consistent. The l a t t e r of these state-ments i s unprovable i n ZFC ( [3], p. 56 ). So i n order to proceed very f a r with the set t h e o r e t i c a l manipulation of the models defined above, i t becomes convenient and some-times necessary to add some new axiom to ZFC which asserts t h e i r existence. [3], p. 78 discusses t h i s . The following i s a r e l a t i v e l y strong form of model existence axiom, and w i l l be adequate for our purposes. Axiom A There i s a set H and a binary r e l a t i o n R well-9 founded on H, such that 3HE = tensional model of ZFC. (H,R) i s an ex-Any t r a n s i t i v e model i s extensional ( see [9], p. 21, and note that t h i s proof i s v a l i d for our d e f i n i t i o n of model ). For any model JHE = (H,R) s a t i s f y i n g Axiom A, the Mostowski Collapsing Theorem ( [9], p. 27 ) guarantees the existence of a unique standard, t r a n s i t i v e model IK = (K, K2iVc) which i s isomorphic to I I . To see that 3K i s t r u l y a model, we must keep i n mind that the isomorphic copy of a set i s also a set ( by the Axiom of Replacement ). This gives us the following more useful form of Axiom A. Axiom A' There i s a set K such that 3K = (K,K2Ae) i s a ( standard ) t r a n s i t i v e model of ZFC. By the Axiom of Regularity we know that K 2Ae i s well-founded ( [3], p. 54 ), and so the two statements A and A' are equivalent. Axiom A implies the existence of a minimal standard t r a n s i t i v e model JM0 of ZFC; one that i s countable and i s a submodel of a l l other standard t r a n s i t i v e models of ZFC ( [3], p. 83; [24], p. 197 ). M 0 has no standard proper submodels which are t r a n s i t i v e . Since M Q does not s a t i s -fy A,-Axiom A cannot be proved from the other axioms of ZFC ( [4], p. 110; c f . [26], p. 83 and [9], p. 37 ). The upward Lowenheim-Skolem Theorem c e r t a i n l y allows 10 us to pick K uncountable. We do t h i s to prevent JK = M 0 . From t h i s point we f i x t h i s JK , writing e for K 2 f i E . . A l l further models we w i l l consider are understood to s a t i s f y the condition that t h e i r universes belong to IK . Because K i s a set, the downward Lowenheim-Skolem Theorem allows us to construct, within ZFC, t r a n s i t i v e submodels of JK ( [3], p. 18, c.f. p. 79, where the problem of models with class universes i s discussed ). By a suitable argu-ment, one such model JM i s countable and has countable rank 1 ( [4], p. 110 ), hence M e K. We f i x JM = (M,M2Ae), and c a l l i t the ground model. By standardness and t r a n s i t i v i t y , ordinals i n IK and 3M are ordinals i n the r e a l sense. The class of ordinals i n IK turns out to be the least ordinal not i n JK ( [24], p. 197 ). We now turn to the topic of Boolean algebra, begin-ning with the d e f i n i t i o n . D e f i n i t i o n 0.2 B i s a Boolean algebra i f B = (B,\u2022,+,-, 0,1) where B i s a set, \u2022 and + are binary operations on B, - i s a unary operation on B, and 0 and 1 are d i s t i n -guished constants i n B, a l l of which s a t i s f y : 1 T r a n s i t i v i t y guarantees the existence of a rank func-t i o n ( [3] , pp. 68-9 ). 11 (a) X + y = y + X , x . y = y \u2022 X \u2022 (b) X + ( y + z ) = ( x + y ) + z X \u2022 ( y \u2022 z ) = ( x . y ) \u2022 z \u2022 (c) ( X + y ) . z = ( x . z ) + ( y \u2022 ( X \u2022 y ) + z = ( X + z ) \u2022 ( y + (d) X + X = X r Y \u2022 y = y \u2022 (e) X + (--x) - 1 , x . (-x) = 0 \u2022 (f) - ( X + y ) = -x . -y - ( X \u2022 y ) = -x + -y \u2022 (g) - (-x) = X \u2022 We note that B i s p a r t i a l l y ordered by the r e l a t i o n : x >^ y i f f x = y + x . With t h i s p a r t i a l order, x + y corresponds with sup(x,y) = i n f [ u : u :>_ x, u >_ y ] , and x-y corresponds with in f (x,y) = sup [ u : u \u00a3 x, u <_ y ]. Generalizing t h i s notion, we write EA for sup (A) and IIA for i n f (A) , when A <= B. I t follows that EB = 1, and TIB = 0. It i s conven-ient to adopt the convention: Z0 = 0 and J10 = 1 . D e f i n i t i o n 0.3 (a) A Boolean algebra B i s complete i f for each A c B, EA and HA e x i s t , and EA s B and IIA e B. (b) A Boolean algebra B i s M-complete i f B e M, and for each A cr B: A e M implies EA e B and IIA e B. 12 There i s no problem regarding the existence of Boolean algebras i n models. Since any f i e l d of sets ( e.g. the power set JP (x) of x ) i s a Boolean algebra, each model abounds i n Boolean algebras. If B i s a Boolean algebra in M, then i t i s clear that B i s a Boolean algebra i n JK . We need not be concerned then, about losing Boolean alge-bras when we move from a given model to a more incl u s i v e model. However, i t i s quite conceivable that an incomplete Boolean algebra exists which i s complete i n a certain model THE simply because the A cr B for which LA \/ B do not belong to H. An JM-complete Boolean algebra i s therefore not necessarily a JK-complete Boolean algebra. For the remainder of t h i s section we w i l l consider B to be a fixed JMC-complete Boolean algebra. Without danger of confusion, we w i l l write B for both B and i t s under-l y i n g set B, and write JMC i n places where i t s universe M i s understood ( e . g . x e JMC ) . Two processes w i l l now be dealt with. The f i r s t i s B the construction of the Boolean-valued model JM from M and B. B The Boolean-valued model JMC e JK may be thought of as a generalization of the ground model JMC, where set theo-r e t i c statements may be evaluated for t h e i r \"degree\" of truth. More precisely, where c l a s s i c a l l o gic allows only two truth values, the Boolean-valued model JMC assigns as truth value a member of the complete ( i n JMC ) Boolean algebra B to each statement. This explains the name Boolean valued model. If B i s the two-elernent algebra de-noted {0,1} , the notion of Boolean-valued model reduces to g the c l a s s i c a l notion of model. We define JMC from within M, by induction on the ordinals less than Q-^-r the least ordinal not i n JMC. De f i n i t i o n 0.4 JMCB = {0} B B M = U JMC0 , i f a i s a l i m i t o r d i n a l . B ^ M a + ^ = { x : x i s a function, dom(x) c B JMC^ , rng(x) c B } M B = U JM B * a < 9 M g Notice that each element x of M must by induction g belong to for some least a. This a we may c a l l the g rank p (x) of that p a r t i c u l a r object i n JMC . Even though we are working within the ground model, g i t does not follow that JMC e JMC, and i n general t h i s i s fa l s e . There i s , however, a concrete way of envisioning g JMC as being inside JMC . This i s done v i a the following v B embedding functor : JMC \u2014>- JMC , defined by t r a n s f i n i t e induction on p(x). V D e f i n i t i o n 0.5 (a) 0 = 0 v B (b) for each x e JMC, we have x e JMC , with dom(x) = { y : y e x }, and x(y) = 1 for each y e x. 14 Notice that p(y) < p(x) i f y e x, so that x i s indeed defined i n terms of elements of lower rank. The v-functor i l l u s t r a t e s each set x e M as a spe-v B c i a l i z e d c h a r a c t e r i s t i c function x e ML\" . Because each V y e x i s also a set i n M, y i s also a c h a r a c t e r i s t i c func-V V t i o n of t h i s type, and so x becomes a composition of charac-t e r i s t i c functions on sets of c h a r a c t e r i s t i c functions. The rank p(x) serves to indicate how long t h i s process has gone on. It i s clear that other function-objects ex i s t i n M whose ranges include values other than 1. These of course have no pre-image by v among the sets of M, but they show that the M construction enables the handling of objects which may be s e t - l i k e to varying degrees. This presents us with the p o s s i b i l i t y of considering some of these ob-jects s e t - l i k e enough to combine with the sets of M, thus forming a new, more i n c l u s i v e model of ZFC. This i s the subject of the l a s t portion of t h i s section. It i s possible now to define a Boolean value ftfjCx^, .. ... ,xn) ] e B for each formula cf> of n free variables, and each x,, ... ,x e M . These Boolean values behave l i k e 1 n the conventional truth values of f i r s t order predicate c a l -culus, but since they belong to the M-complete Boolean algebra B, thay extend our notion of semantics beyond the usual d u a l i t y of truth (1) and f a l s i t y (0). The Boolean values I x e y ] and j x = y ] are defined for x, y e M by t r a n s f i n i t e induction on the lexicograph-i c a l ordering ( p(x), p(y) ) ( i . e . the ordering defined by: (a,3) > (S,y) i f f a > S or a = 6, 3 > Y )\u2022 D e f i n i t i o n 0.6 (a) [ x e y I = (b) II x = y I = z ( y ( z ) - i z = x ] ) , z e dom(y) n ( - x ( z ) + I z \" \u00a3 y 1 ) z e dom(x) \u2022 n ( -y(z) + I z e x ] ) . z e dom(y) For a discussion of the form and e f f i c a c y of the above and similar d e f i n i t i o n s , see [17], pp. 41 - 4, and [25], pp. 121 - 2. Having defined H cj> J for cj) an atomic formula, we ex-tend the d e f i n i t i o n to include any set theoretic formula. D e f i n i t i o n 0.7 (a) (b) (c) (d) (e) I i * 1 = [ Cf> & ty ] I * v ty ]] II * - IP 1 I (vx)4> i| I * 1 1 cj) l-l ty I i (j) i + i ty i -I \u2022 1 + I iM n B l (x) ] x e M (f) I (ax)cf> ]] = Z B E cf)(x) TJ X \u00a3 M The M-completeness of B ensures that 0.6 (a), (b) and 0.7 (e), (f) are well-defined. The following i s a t r i v i a l but useful consequence of the above d e f i n i t i o n s . 16 Lemma 0.8 I a) J < I n> ] i f f [ <|> ->- u> ]] = 1 A series of lemmas now follows, which give several useful r e l a t i o n s concerning the Boolean values of some s p e c i f i c formulas. The proofs are straightforward, and are usually accomplished by t r a n s f i n i t e induction on ( p(x ) , p(y) ). A sample proof accompanies the f i r s t of these lemmas. Lemma 0.9 (a) I x = x ] = 1 . (b) x(y) < I y e x 1 (c) [ x = y J = il y = x ] . Proof: Suppose (a) to be true for any x s a t i s f y i n g p(x) < y . Let p(x) = y now. By d e f i n i t i o n : (i) [ x\" = x 1 = II (-x(y) + I y e x I ) . y e dom(x) For each y e dom(x): ( i i ) E y e x ] = \u00a3 ( x (u) -I u = y 1 ) u e dom(x) > x(y) \u2022 ily = y ] = x(y) \u2022 1, by hypothesis, as p(y) < y . Since x(y) i s defined only where y e dom(x), (b) follows. Substituting for [ y e x ] i n ( i ) , we have: K x = x ] 1 n (-x(z) + x(z) ) = 1 . z e dom(x) Therefore: x = x = 1 for p(x) = y . Calculating d i r e c t l y : ii 0 = 0 1 = 1 ( the Boolean infimum of the empty c o l l e c t i o n i s 1 ), so (a) follows by t r a n s f i n i t e induction. (c) i s v e r i f i e d d i r e c t l y from the d e f i n i t i o n . The next three interdependent statements are proved simultaneously by t r a n s f i n i t e induction on ( p(x ) , p(y) ) ( see [25], p. 123 ). Lemma 0.10 (a) | x = y l ' I x e z J ^ I y e z l (b) t x e z 1 \u2022 I z = y I < I x E y I . (c) I x = y ] \u2022 I y = z 1 < I x = z } . The lemma below follows by induction on the complexity of cj), using the fact that the previous lemma establishes the r e s u l t for atomic formulas. Lemma 0.11 I x = y 1 \u2022 II a> (x) II < I (y) J . Lemma 0.12 (a) [[ ( ay e x)f(y) 1 = E (x(y)\u00abtt o) (y) 11) y e dom(x) (b) I (Yy e x)tj>(y) 1 = n (x(y)-[ o) (y) ] ) y e dom(x) Proof: See Def i n i t i o n 0.13 [25], p. 125 . g Let x, , ... ,x e M . ** y, F ( 3 ) = F ( y ) . 20 Lemma 0.18 For each formula cf> of set theory, the function F (a ) = 1 \u00a3(}) (x)J X \u00a3 M a i s nondecreasing and eventually constant. Proof: For increasingly larger ordinals 3 , F (3) i s a Boolean supremum taken over increasingly g larger sets JMQ , hence F i s obviously non-\u00bb p decreasing. We define a function H: B *\u2022 0__ by: H(a) = in f [ 3 : a < F (3) ] . Since B e M, an ordinal y exists which i s the supremum of the image of B by H. For each 3 > y , F (3) = F (y) . The fact that B i s a set i s c r u c i a l ; neither the Axiom of Replacement ( 0.19 ), nor the Axiom of Power Set ( 0.21 ), g nor the Maximum P r i n c i p l e ( 0.2 6 ) hold i n some 3M where B i s a proper class ( [25], p. 196 ). Lemma 0.19 For each formula o) of set theory: M B (=(Vx) (3y) (Vu e x) [ (3v)cj>(u,v) + ' (3v e y) cj) (u,v) ] Proof: Let x e ]MCB. The function F (3) = \u00a3 D Io>(ufv)l v e 3MC D p i s eventually,constant, by 0.18, for each u \u00a3 dom(x). This enables us to define the function: g(u) = i n f [ a : V3 > a , F (3) = F u ( a ) 1 f o r each u \u00a3 dom(x). We may write: 21 Z B.I<|>(u,v)] \u2022= . 2 E (ufv)]I V \u00a3 3MC V e M . . g (u) B B Therefore, for each x e M there exists y \u00a3 M s a t i s f y i n g : | (Yu \u00a3 x) [ (3v)cf)(u,v) \u2022* (3v \u00a3 y)cj)(u,v) ] 1 n [ -x(u) + n(av)(j)(u,v) ^ (av E y), there i s a set g y e ME s a t i s f y i n g : dom(y) = dom(x) , and: I ' (Vz e y) ( z e x & o>(z) ) I = 1 , II (V Z e x) ( <|>(z) -> z e y ) 1 \u2022 = 1 . The main application of the above lemma i s i n the next r e s u l t . g Lemma 0.21 M \\= (Vx) (3y) (Vu) ( u cz x ->\u2022 u e y ) B B Proof: Let x, u e M . From 0.20, there exists v e M sa t i s f y i n g : dom(v) = dom(x) , I v = u A x ] =1. For each t edom(x) we have: [ t e v j = [ t e u l - i t E x ] , thus, I t E v l < I t e x l g Following 0.17, we define y e M as follows: dom(y) = {' z : dom(z) = dom(x), t e dom(x) I t E z l < I t e x l } , and y(z) = 1 for z e dom(y). We know that y ^ 0 when x ^ 0, since v e dom(y). Furthermore: I u c x l = I u c x l ' [ v = u ( l x I < I u = v l , so that: I u c x I < I I u = z I = I u e y 1 . z e dom(y) OH the other hand: 23 1 u e Y 1 = I tt z = u H z e dom(y) (i), = I I z = u M z c x l , z E dom(y) since z \u00a3 dom(y) implies I z c x 1 = 1 0.11 gives (i) < Z ttucxl=lucxl. z \u00a3 dom(y) B B Given any x E M , we have produced a y \u00a3 M s a t i s f y i n g I u c x 1 = |[ u e y I , for each g u \u00a3 M . The r e s u l t now follows. Lemma 0.21 establishes the v a l i d i t y of the Axiom of g Power Set i n 3MC . The Axiom of I n f i n i t y may be validated by various s t r a -tegies. Jech sketches a recursive construction of an i n -g f i n i t e set i n M ( [9], p. 56 ). Requiring s l i g h t l y more background i s Rosser's proof that A w i s i n f i n i t e 1 = 1 ( [17], p. 77 ). At t h i s point we quote a general theorem that y i e l d s the immediate v a l i d a t i o n of the Axiom of I n f i n -i t y , as well as that of the Null Set Axiom. (j)(x^, \u2022\u2022\u2022 ' x n ) ^ s a bounded formula of set theory i f each of the quantified variables of 4> are r e s t r i c t e d to one of the sets x^, . . . ,x n , e.g. (Vx \u00a3 a) (3y \u00a3 b) ( x e y ) . Theorem 0.22 If ****) (x e y) (b) M B (= (Vx e 0) ( x ? x ) . The remaining lemmas of t h i s series culminate i n the v a l i d i t y proof of the f i n a l axiom of ZFC, namely the Axiom of Choice. D e f i n i t i o n 0.24 Let u e B and u f 0 . { u\u201e: 6 e I } i s p a p a r t i t i o n of u i f \u00a3 u R = u , and B e l u y \u2022 u 6 = 0 for y it 5 ( I t= 0 M , I e M) . Lemma 0.2 5 Let ( u Q : B e I ) be a p a r t i t i o n of u e B, p u ^ 0. Let { t'g : B e l l e M B, and 1 <= Q m ' g I e M. Then there exists t e M such that: u Q < II t = t Q ]) for each g e l . P \u2014 p Proof: Letting a = sup P(t\u201e) , we define t as follows: B e l p dom(t) = M a + 1 t(z) = \u00a3 u * t Q ( z ) Pel B B An immediate consequence i s that for each 3 e I g and each z e M a + 1 : u^'t^(z) = u^'t(z) This fact gives us two c a l c u l a t i o n s : Ci) Ug - ( -t(z) + IT z e t 6 1 ) > [u 3- -t(z) ] + [u e\u00bbt g(z) ] = [up\u00bb - t ( z ) ] + [ug\u00bb t(z)] ( i i ) Ug ' C-tgtzi + H z e t 1) ' > [ u g * ~ t g ( z ) ] + [u -t(z) ] = [ V - t 3 ( z ) ] + [ v t 3 ( z ) ] = u 6 . Employing (i) and ( i i ) , we conclude: ff t = t f l I- > u \u2022[[ t = t e 1 p \u2014 P p > ufl- n u \u00ab(-t(z) + ttz e t f l I ) ~ 6 z \u00a3 3MCB 3 3 n u \u2022 ( - t (z)+E z e t 1) z e dom(tg) p p g In s a t i s f y i n g the above lemma M i s said to be a com-plete Boolean-valued structure ( see [25], p. 62 ). The t i n Lemma 0.2 5 i s unique in the sense that i f t and t\" both s a t i s f y the Lemma, then LI t = t 1 I = 1 The next lemma i s known as the Maximum P r i n c i p l e as g i t states that Boolean suprema i n JMC are i n fact maxima. Lemma 0.26 For each formula cj) (x) of set theory there exists t e 3MEB such that II (ax) cf> (x) J = II (t) I Proof: Let u = C ( a x ) (x) I = I Q I $ (x) 1 x \u00a3 M Without loss of generality, we assume u ^ 0 . It follows that for some t Q we have: u 0 = I cj)(t0) II > 0 . g The sequence { t g : 3 < a } < = M i s constructed inductively. If u- II -u > 0 , we may B Y < 3 pick t Q \u00a3 M such that: P o < u f i = I (x,y) be: (Vz)[ ( z e x & z ^ 0 ) - > - ( a t ) ( t e z & ( z , t ) B e y ) ] . Using Lemma 0.30 and others, we calculate: I *(x,y) II _> n [ -x(z) + ( -sup(z)+x(z) \u2022 Z E M B sup(z))] = 1 . Now we w i l l show II y i s a function B = 1 . Actually, only part (c) of D e f i n i t i o n 0.29 29 needs demonstration. Let g be the function i n M defined by: dom(g) = dom(x) ; g(z) = t , i. e . g i s the choice function on dom(x) described above. g i s extensional, i . e . (i) Vz e dom(x), I z = z 1 I < II g(z) = g ( z ' ) II . The v e r i f i c a t i o n of the above involves an elementary applicaton of 0.9 (b) and 0.11 . g For each z e dom(x) and t e M , we have: [ ( z , t ) B e y 1 = 1 y( (z\u00ab ,g(z') ) B ) - | ( z , t ) B = (z ' , g (z ')) B l D z*edom(x) = Z x ( z ' ) \u2022[[ (z,t) = ( z ' ^ f z 1 ) ! (by 0.30) z 'e dom (x) = T. x ( z ' ) \u2022 Ez = z ' ]\u2022[ t = g ( z ' ) 1 z 'e dom (x) < I x ( z ' ) - [ g ( z ) = g ( z ' ) ] - I[t = g ( z')l z'edom(x) ( by (i) above ) < E g(z) = t ] . Applying t h i s c a l c u l a t i o n , we conclude: I ( z , t ) B e yl \u2022 I ( z , f ) B e yl_ x implies y e G . (iv) Vx e B , x \u00a3 G o r - x \u00a3 G . (b) G c B i s an M-generic u l t r a f i l t e r i f , i n addition to the above, G s a t i s f i e s : (v) A cr G, A \u00a3 3MC implies IIA \u00a3 G . G i s just a f i l t e r i f i t s a t i s f i e s (i) - ( i i i ) above. G i s a proper f i l t e r i f G ^ B. Condition (iv) i s equivalent to saying that G i s a maximal proper f i l t e r , i . e . one that i s not properly included i n any other proper f i l t e r . A useful equivalent to 0.32 i s the following. Lemma 0.33 An u l t r a f i l t e r G on B i s M-generic i f f for each p a r t i t i o n A of u e G such that A \u00a3 M, there exists b e B such that A fi G = {b} . Proof: Let A c B , A e M, then G i s M-generic i f f : (i) IIA \u00a3 G implies A j-\u00a3 G . We write: A 1 = { -a: a \u00a3 A } Since G i s an u l t r a f i l t e r , we have: IIA jzf G i f f -(IIA) e G i f f \u00a3A* e G S i m i l a r l y , A \u00a3 G i f f (3a \u00a3A)( a \u00a3 G ) i f f (3aeA)(-aeG) i f f (3aeA')( a e G) Hence (i) i s equivalent to: ( i i ) EA' e G implies (3aeA')( a e G) Given A 1, by simply taking the supremum of the a's which s a t i s f y ( i i ) , we arrive at a unique b eA' fi G , with no e s s e n t i a l change i n A'. D e f i n i t i o n 0.34 Let G be an M-generic u l t r a f i l t e r on B. By t r a n s f i n i t e induction on p(x), we define the interpretation functor i\u201e of M by G: (a) i G ( 0 ) = 0 . (b) i Q ( x ) =' { i G ( y ) : x(y) \u00a3 G } . We usually write ' i ' for i . , , dropping reference to G when i t i s understood. D e f i n i t i o n 0.35 M [G] = { i (x) : x e M B } i s c a l l e d the generic extension of 3MC by G, where G i s an ME-generic u l t r a f i l t e r on B. As seen below, the notation M[G] suggests ( as i n f i e l d theory ) what i t should. Theorem 0.36 M[G] i s the least model of ZFC exten-ding M and containing (G} The s i t u a t i o n i s summarized i n the commutative diagram below: inclusion G At t h i s point we w i l l only show part of 0.36, i . e . that M[G] i s a model of ZFC extending M and containing G as an element. This w i l l be done i n the next series of lemmas, ending with 0.40 . The minimality of ME [G] i s es-s e n t i a l l y a consequence of our Lemma 5.5 . [9], p. 59 gives another proof using absoluteness, which would be almost un-changed i n our system of models. Lemma 0.37 For each x e W, i (x) = x . V Proof: Induction on p(x) : i(0) = i(0) = 0 . i(x) ='.{ i(y) : x(y) e G } = { y : x(y) e G } , as p (y) < p(x) , = { y : y e x } , as x(y) = 1 \u00a3 G , g We define the canonical generic u l t r a f l i t e r G on MI : V V dom(G) = { x : x E B } ; G(x) = x , Vx \u00a3 B G belongs to M by d e f i n i t i o n . Lemma 0.38 G \u00a3 M[G] Proof: i(G) = { i (x) : G(x) \u00a3 G } = { i (x) : x \u00a3 G } = G . Suppose x \u00a3 M , x \u00a3 ML\" [G] , and i (x) = x . Then we V say that x i s a name for x. For example, x i s a name for x. and G i s a name for G. Lemma 0.39 If x, y are names for x, y \u00a3 ME [G], respectively then: X \u00a3 y i f f [ [ x e y f l e G , and x = y i f f I x = y 1 \u00a3 G . Proof: (a) Given that x \u00a3 y, we show I x \u00a3 y I \u00a3 G. If x \u00a3 y, then there exists z Q \u00a3 dom(y) such 35 that y_(z0.)_ e G and i ( z 0 ) = x . Proceeding by t r a n s f i n i t e induction on ( p (x) , p (y) ), we assume by induction hypothesis that I z Q = x 1 eG, as p(z 0) p (y) . Hence y ( z 0 ) * l z Q = x 1 e Gf and since II x e y 1 > y (z Q) \u2022 I z Q = x 1 , we have that II x e y 1 E G . (b) For the converse of (a), see [9], p. 58 . (c) Given that x = y, we show II x = y I e G. Since i (y) = { i(z) : y(z) e G } = ( i ( z ) : x ( z ) e G ^ = i ( ^ ' we have: Vz e M B, y(z) e G i f f x(z) e G . Hence, for a l l z e dom(x): (i) ( x(z) \u00a3 G ).+ (-x(z) e G ) + (-x (z) +11 z e y l e G ) , ( i i ) ( x(z) e G ) -> ( i ( z ) e y ) ^ ( I z e y J e G ) ( as p(z) < p (x) ) + C-x(z) + [ z e y ] e G ) . Si m i l a r l y , for a l l z e dom(y): ( i i i ) (y(z) \u00a3 G ) -> (-y(z) + II z e x ]] e G ) , (iv) ( y(z) e G ) (-y(z) +11 z e x l e G ) . From the d e f i n i t i o n of II x = y H , (i) - (iv) above, and the genericity of G ( 0.32 ), i t follows that I x = y I e G. (d) The converse of (c). Given that fl x =' y TJ e G; for a l l z e dom (x) : x(z) e G implies _j z e y ]] e G, because G i s an u l t r a f i l t e r and -x(z) + [[ z e y JJ e G. So, i f x ( z ) e G ( i . e . i(z) e x ) then tt z e y 1 e G, and by induction hypothesis i ( z ) e y, as p(z) < p(x) . For the same reason, for a l l z e dom(y): y(z) e G ( i . e . i( z ) e y ) implies [[ z e x J e G, which by induction hypothesis implies i ( z ) e x, as p(z) < p (y) . We conclude that for a l l z e dom(x) u dom(y) such that x(z) e G and y(z) e G: i(z) e x i f f i(z) e y Hence, x = y . Lemma 0.40 If x^, ... ,x n e M are names for x^, ... ,x n e M [G] , and cj) i s a formula of set theory, then: M[G] H <$>{xir ... ,x n) i f f ff1(l)(x1, ... ,x n) H e G. Proof: This follows from the previous lemma by induc-ti o n on the complexity of cf> . 0.16 and 0.40 prove that M[G] i s a model of ZFC. 0.39 implies that M[G] i s standard, and i t i s not hard to show ( using 0.40 ) that M[G] i s t r a n s i t i v e . Because i Q i s defined by t r a n s f i n i t e induction over 0 e 3K , we have 37 i\u201e e IK, The Axiom of Replacement thus implies that M [G] e K . The other d e t a i l s of the diagram on p. 33 follow from 0.37 and 0.38 . Since 3K i s t r a n s i t i v e , our diagram seems to indicate that G e 3K . A l l along however, our t a c i t assumption has been that M-generic u l t r a f i l t e r s do e x i s t . To make such an assumption i s equivalent to adding a very strong axiom to ZFC. Martin's Axiom, which i s a weaker and more reason-able form of t h i s assumption, may be invoked for t h i s pur-pose ( see [13] ), but we would l i k e to avoid any further additions to our foundations. We s h a l l now show that the foundations l a i d at the beginning of t h i s section are enough to provide the existence of an M-generic u l t r a f i l t e r G i n 3K . An u l t r a f i l t e r H a i s said to be p r i n c i p a l i f i t i s of the form { b e B : b > a }. Suppose that H i s M-generic. \u2014 c l Then 0.33 implies that there are no p a r t i t i o n s of a e B i n M. Within M, a i s an atom, or minimal non-zero member of B ( even i f B i s i n r e a l i t y nonatomic, i . e . having no atoms ). A sim i l a r argument shows that i f an M-generic u l t r a f i l t e r G belongs to M, then TIG i s an atom of B. Since we do not r e s t r i c t our attention to Boolean algebras having atoms i n the sequel, we cannot rule out the p o s s i b i l i t y that each M-generic u l t r a f i l t e r G on B i s non-principal and that G \u00a3 M. G, i f i t exists at a l l , may be highly non-constructive . 38 D e f i n i t i o n 0.41 Let F be a family of subsets of a Boolean algebra B. A f i l t e r U on B i s F-complete i f for each E e F such that HE e B: E cr TJ implies HE e U . There i s an obvious redundancy i n the above d e f i n i t i o n i f B i s complete. Two elements a, b e B are compatible i f a*b ^ 0 . A pairwise compatible subset of B i s one whose members are compatible with each other. It i s apparent that f i l t e r s are pairwise compatible, and that each subset of a pairwise compatible set i s pairwise compatible. Below, we have a t r i v i a l extension lemma which w i l l be helpful i n construc-ting and extending pairwise compatible sets. Lemma 0.42 If H c B i s pairwise compatible, then for each b e B, either H U {b} or H U {-b} i s pairwise compatible. If H i s a pairwise compatible subset of B, then we can enlarge i t to the following f i l t e r : J = { z e B : z > II a, , a , e H l . k < n By maximalization, we may further extend J to an u l t r a f i l -ter. This i s expressed i n the well-known U l t r a f i l t e r Theorem below. Theorem 0.43 Each pairwise compatible subset of B i s contained i n some u l t r a f i l t e r on B. Since the above theorem follows from AC, i t holds i n IK , and u l t r a f i l t e r s are p l e n t i f u l in IK . The d i f f i c u l t y of the existence problem we are considering must l i e i n the property of genericity. Theorem 0.44 Given a countable family F of subsets of a Boolean algebra B, and an F-com-plete f i l t e r G Q on B, there exists an F-complete u l t r a f i l t e r G on B extend-ing G 0 . Proof: Let F* = { A \u00a3 F : - n A \u00a3 G0 } . Since F i s countable, we may enumerate F* = { A D, ... ,A , ... } , and define P n n II (A ) . By d e f i n i t i o n , P ^ 0 , for n n each n. F* has the following property: Vn, Va e G D , a-p n ? 0 , since a\u00abp =0 implies a < -p , but ^n \u2014 cn -P i G< ,Q . Because of t h i s , we know that H D = G oy'{p 0} i s pairwise compat-i b l e . Having defined H^ and assuming that i t i s pairwise compatible, we use 0.42 to define H ,, as H U (p } t i f n+1 n *n th i s i s pairwise compatible; or HfiU i ~ P n ^ otherwise. H = y H i s thus a pairwise n n compatible set containing G Q . We ex-tend H to an u l t r a f i l t e r G by way of 0.43, 40 G i s c l e a r l y G--complete. Theorem 0.44 i s an extended form of the Rasiowa-Sikorski Lemma ( c f . [25], p. 29 et seq. ). While the usual form of t h i s r e s u l t i s s u f f i c i e n t for our present purpose, we w i l l need the stronger hypothesis of 0.4 4 in Section 4. It may seem natural to release F from the coun t a b i l i t y re-s t r i c t i o n , but t h i s cannot be done without making further r e s t r i c t i o n s on B ( e.g. Martin's Axiom [9], p. 99 ). I t now becomes apparent just why we have picked a countable ground model M. Corollary 0.45 If M <=\u2022 IK i s a countable model of ZFC, and B i s a complete Boolean algebra i n M, then M-generic u l t r a f i l t e r s on B exi s t i n IK . M Proof: G i s M-generic i f f G i s IP (B)-complete. Since M nP (B) i s countable, Theorem 0.44 y i e l d s the r e s u l t . This concludes our study of M and M[G] as abstract objects. In the sequel we w i l l work with p a r t i c u l a r ex-amples of these constructions, and v i r t u a l l y a l l of the material i n t h i s section w i l l f i n d application. By now we are acquainted with the use of a generic u l t -r a f i l t e r as a type of decision process capable of evaluating any set theoretic statement regarding i t s v a l i d i t y i n M [G]\u201e . In practice, many of, the properties of M{GJ w i l l be demon-strated by Boolean-valued calculations involving generic u l t r a f i l t e r s . We w i l l also f i n d i n the sequel that generic u l t r a f i l t e r s can be used to r e f l e c t a number of algebraic and a n a l y t i c a l notions. They can be used i n certain s i t u -ations to provide d i r e c t answers to purely non-logical problems. 42 Section 1 : The Lebesgue Measure Problem and the Axiom of Choice The Axioms of Pairing, Unions, and I n f i n i t y ensure that a l l standard t r a n s i t i v e models of ZF contain the set of natural numbers w ( see [24],p.129 ). Well-known methods ( e.g. Dedekind cuts ) of generating the r a t i o n a l and r e a l numbers are e a s i l y duplicated i n these models ( see [16],p.271 ), however the set of Dedekind cuts gen-erated from to may vary from model to model. It i s possible thus to express statements of r e a l analysis as formulas i n ZF. As the concept of Lebesgue measure i s definable in t h i s context, we may view the Lebesgue Measure Problem from the vantage point of ZF by asking whether models of ZF e x i s t which s a t i s f y the statement: LM A l l sets of reals are Lebesgue measureable. In the event that a model IE does exi s t such that IE |= LM , we immediately conclude that IE ^ AC , for AC-*--] LM . For an analyst, our model 3E might be unattractive as there i s no guarantee that certain basic prerequisites of analysis hold on IE . Not only are the non-constructive p r i n c i p l e s , such as the Hahn-Banach Theorem, derived from some form of AC, but so are some commonplace facts l i k e the re g u l a r i t y of (. countable unions of countable sets are countable ) . Moreover, the presence of AC in some ( possibly weaker ) form i s necessary to provide acceptable properties for 43 Lebesgue measure in JE , There are two main characterizations of the basic top-o l o g i c a l notions of metric spaces: (a) e - < 5 c r i t e r i a . (b) sequential l i m i t c r i t e r i a . The d e f i n i t i o n s of a l i m i t point and the closure of a set, as well as continuity of functions have obvious versions i n (a) and (b). If our model under discussion s a t i s f i e s the Heine-Borel Theorem ( which may not be the case; see 1.2 ), then versions in (a) and (b) e x i s t for the d e f i n i -t i o n of compactness of a set. Using AC we can prove the equivalence of both versions of a l l the d e f i n i t i o n s men-tioned above. What happens i f our model does not s a t i s f y AC ? Proposition 1.1 For each of the following notions there i s a model of ZF not s a t i s f y i n g AC i n which versions (a) and (b) of said no-t i o n are not equivalent. (i) l i m i t point of a set. ( i i ) closure of a set. ( i i i ) continuity of a function. (iv) compactness of a set. Proposition 1.2 For each of the following statements there i s a model of ZF not s a t i s f y i n g AC i n which said statement holds: Ci) co^ i s singular. ( i i ) the set of reals IR i s the union of countably many countable sets. ( i i i ) there i s an i n f i n i t e set of reals having no countable subset. (iv) there i s a subspace of the reals which i s not separable. (v) the Heine-Borel Theorem i s f a l s e . See [10], pp. 141 - 4 for a demonstration of the above r e s u l t s . As far as the needs of the analysis and topology of the reals are concerned, the appropriate weakening of AC i s the following statement, known as the Countable Axiom of Choice. AC^ Every countable c o l l e c t i o n of nonempty sets has a choice function. If IE |= ZF + AC^ then versions (a) and (b) of each of (i) - (iv) i n Proposition 1.1 are equivalent i n IE . . Also, none of the statements (i) - (v) i n Proposition 1.2 can hold i n any model of AC^ . In fact, we have the following r e s u l t . Proposition 1.3 If IE f= ZF + AC^ then each of the f o l l -owing statements must hold i n IE : (a) the Heine-Borel Theorem (\"h)_ every subspace of a separable metric space i s separable. (c) Lebesgue measure exists and i s count-tably additive. (d) the family of F i r s t Category sets i s countable additive. Proof: See [10], p. 21 - 2, p. 29 . The Baire Category Theorem does not depend at a l l on AC. AC^ does not imply the f u l l strength of the general Hahn-Banach Theorem; the McAloon model of Section 6 v e r i f i e s t h i s ( see [23], pp. 2 - 3 ). However, AC^ e a s i l y y i e l d s the Hahn-Banach Theorem for separable Banach spaces. Since the family of Borel sets w i l l have a special significance i n l a t e r constructions, i t i s worthwhile to examine the r e l a t i o n i t has with AC. There are two usual d e f i n i t i o n s for t h i s family, IB . D e f i n i t i o n 1.4 (a) IB i s the smallest a-algebra of sets of reals containing the open sets, (b) IB i s the c o l l e c t i o n of sets hyper-arithmetic i n some re a l ( see [21], p. 179 ); or, by a theorem of Souslin: IB i s the c o l l e c t i o n of A. sets i n the Projective Hierarchy ( see [21], p. 185, c f . [11], v . l , p. 453, et seq. ). As (b) i s the type of d e f i n i t i o n we w i l l r e l y on, we \u2022 need AC at l e a s t , i n order to show tha t IB i s c l o s e d under co \u2022 countable unions (. we must p i c k a code f o r each of count-ably many Bore l sets ). Without AC, (a) and (b) above are not g e n e r a l l y e q u i v a l e n t ; however AC^ i s strong enough to guarantee the equivalence of (a) and (b), and show ( see [10], P. 22 ): (i) IB = u IB a where B 0 the open * a i s IT i i f : cj> \u00ab-\u00bb- ( Vx l f ... ,x n e oow )ty , where ty i s a formula whose only quantifiers are of the form Vy e w , or ay e oo . The following i s a syntactic c l a s s i f i c a t i o n of the formulas of ZF, known as the Levy hierarchy ( see [12] ). D e f i n i t i o n 2.2 (a) A formula i s E Q = n o i f i t i s bounded ( see p. 21 ). (b) ty i s \u00a3 n + 1 i f cj) = 3xty where ty i s n . n (c) cj) i s n i f cj) = Yxip where ty i s E . n (d) cj> i s Z* F, resp. II^ F i f ZF |- ty \u00ab-*\u2022 ty where ty i s E , resp. II r n n 48 ZF (e) o) i s A , resp. A i f d> i s both n n Y \u00a3 and II , resp. A and II n n ' c n n Let IE , IF be standard models of ZF with universes E, F respectively. It i s obvious that IE i s a submodel ( see [1] , p. 21 ) of IF i f E cr F . For such standard models s a t i s f y i n g t h i s condition we write IE cr IF Let cj> be a formula of ZF. An assignment f of a) i n IE i s a mapping of the free variables of a> into E; we write 4>[f] for the sub-s t i t u t i o n of f (x) for each free variable x occuring i n tf>. We now define the fundamental model theoretic concept of t h i s section, the notion of absoluteness between stan-dard t r a n s i t i v e models of ZF. There are many notions of absoluteness i n the l i t e r a t u r e . Unlike the absoluteness of Goedel ( see [7] ) or of Cohen ( see [3] ), the d e f i n i -t i o n we use ( due to Shoenfield; see [4], pp. 85, 106 ) does not employ r e l a t i v i z a t i o n of formulas to t r a n s i t i v e classes. D e f i n i t i o n 2.3 Let IE , IF be standard t r a n s i t i v e models of ZF, and l e t IE cr IF (a) A formula i s absolute between IE and IF i f f for a l l assignments f of a) i n IE : IE cf>[f] i f f IF f= (x,y) i s preserved under extension from IE to IF , by def-i n i t i o n of f= . ZF Hence \u00a3^ formulas are preserved under -the above extension ( i . e . they hold in IF i f they hold i n IE ) . Sim i l a r l y , Vx 0, s n i s nonempty and has length k, say. Let be the i n i t i a l segment of s n having length k-1. Let the f i n a l segment of s be n, \u201e. Then n* 0 , *(a,n*)(0) = 0 (mod 3) . (c) $ (a,n*) ( J ( n k , i ) ) ; n\u00bb 0 , $(a,n*)(0) = 1 (mod 3) . (d) $ (a,n*) (i+1) ; n > 0 , (e) $(a,n*)(0) = 2 (mod 3) . The purpose of t h i s function i s to 'decode' a , y i e l d -denotes the concatenation of f i n i t e sequences, 54 ing the codes of the component Bor e l sets from which B^ may be constructed. Let 3 e O J w . We def i n e 3 e coW v i a the f i n i t e sequence s^, > = ( 3(0), ... , 3 ( n - l ) ) . The p v n) f o l l o w i n g Lemmas are due to Solovay. Lemma 2.11 Define cj^Ca) as (V3eoja)) (Sneco) [$ ( a , 3 (n) ) = 0] . Then: IE \\= ^ (a) \u2022\u00ab->\u2022 ( a codes a Bo r e l set ) . I f a codes a Bore l set and x e IR , d e f i n e : 'l ; x e B, Y ( i ) = $(a,i) 0 ; otherwise The previous lemma guarantees the existence of B^ ^ a ^ Lemma 2.12 There i s an a r i t h m e t i c formula ( see [22], p. 160 ) cf>4(a,3,x) such that cj>4(a,3,x) \u2022\u00ab-*- 3=Y Lemma 2.13 Let x e IR . There are formulas ^ ( o ^ x ) . <^^(a,x) such t h a t : (a) IE \\= cj>2(a,x) +*\u2022 ( a codes a Bo r e l set and x e B ) . a (b) IE |= ((^(a\/x) -<-\u00bb\u2022 ( a codes a Bo r e l set and x t Ba ) . Proof: (a) Define fy^ ( a\/ x) a s t n e f o l l o w i n g : (V3ew W) (aS4 (a , 3,x) + 3(0) = 1 ) & ^ ( a ) . o>2(a,x) i f f y(0) = 1 , by Lemma 10, and Y(0) = i f f x e B, , i f f x e B , as $(a,0) = a . \u00b1 Ca) . ***3 (a,x) i f f yCO). = 0 , by Lemma 10, and yCO) = 0 i f f x ft B, , i f f x \u00a3 B . \u2022 *~~ (a, 0) a Both of the above formulas are . Corollary 2.14 There are formulas cf)^(a,3) and _(a,3) as <}>, (a) & <(>_ ( 3 ) & (Vx \u00a3 ]R) ( $ (a,x) v <|> (3,x) ) . Define (..(a,3) & $^(8,a) . Quantifying over the reals i s permissible in the d e f i n i t i o n of _(a,3) . We could o code each r e a l by i t s binary expansion, thus ensuring that 1; i s a code for both ClB. i n IE , and (?I#B. i n IF . By d e f i n i t i o n : # O B . = 0#B. , i i J l i i . i and (b) follows for the case of i n f i n i t e i n t e r -section. Codes for complementation and i n f i n i t e unions are covered i n D e f i n i t i o n 2.8 (b) - (c). 58 Thus (bl. (a) follows from (b). Theorem 2.15 Ccl - Cdl imply Cc) . Of the numerous topological notions that are #-absolute, the two most basic are a l l we need here. Lemma 2.20 (a) ( B i s open ) i s #-absolute. (b) ( B i s closed ) i s #-absolute. Proof: Let {[r. ,r. ]} comprise the closed r a t i o n a l -1k Dk K endpoint i n t e r v a l s containing B. Define y as: Y(0) =2 (mod 3); Y (1) = 1 (mod 3) ; Y(J(k,0)+l) = 2 (mod 3); Y(J(k,l)+l) = 0 (mod 3); Y(J(k,2)+l) = i k ; Y(J(k,3)+l) = j k ; then Y codes the closure of B. B i s closed i f B = B = B^ . (b) follows from Lemma 2.19 . B i s open i f IRxB i s closed. (a) follows from (b) . With si m i l a r arguments ( [23], p. 30 ) we can show that the int e r v a l s are r a t i o n a l over IE . Lemma 2.21 Let a, b e^R.^, then #(a,b) = (a,b); #[a,b] = [a,b]; #{a} = {a} . We now have a l l the tools necessary to explore Lebesgue measure i n t h i s model theoretic setting. Our concept of Lebesgue measure y i s that of an outer measure r OO 0 0 \u2022, y*(E) = i n f { E_(b - a ) : U . (a ,b ) => E , b > a } 1 n=0 n n n=0 n n n n J r e s t r i c t e d to the a-algebra of measurable s e t s . Lemma 2.22 I f IE |= AC then f o r each Lebesgue measurable s e t E i n IE there are s e t s G a n d N such t h a t IE f= E = G-N & y*(N) Proof: We take E measurable t o mean t h a t f o r each E > 0 th e r e i s an open s e t 0 and a c l o s e d s e t F such t h a t F c E c 0 ,and y * ( 0 ^ F ) < e .. \u00a3 \u00a3 \u00a3 E \u00a3 By AC we may p i c k such a p a i r (0, , , F., , ) J oo x\/n 1\/n f o r each n \u00a3 oo . L e t G = n 0, . . Then de-n 1\/n f i n e N = G-E . V n e oo, y* (N) < ^ ^ \/ y \\ Y \\ \/ r ? < 1\/n . We now look a t v a r i o u s cases o f #-absoluteness f o r Lebesgue measure. Lemma 2.23 Let B be a ^ - s e t i n IE , then y E (B) = y ^ (#B) . Pro o f : The e q u a l i t y we wish to prove i s expressed i n a 3K , so our p o i n t o f view i s t h a t o f the diagram f o l l o w i n g C o r o l l a r y 2.17 . (a) Suppose B = y (a \u00bbt>m) , then from Lemmas m=l m 2.19, 2.20, 2.21, the d e f i n i t i o n o f Lebesgue measure, and the n a t u r a l i t y of #, n u (B) = \u00a3, (b - a ) i s #-absolute, and H m=l \u2022m m y E (B) = mw (#B) i n IK . (b) L e t B be any open s e t . Enumerate the s e t s 60 of form i n (a) above: { A n} . Then: V (B) = sup { M (A n) : A n cr. B. } n i s sup of a countable c o l l e c t i o n of #-absolute r e a l s , hence i t i s #-absolute. Cc) L e t B e . We need only l o o k at the r e p r e s e c t a t i o n : B = n 0 ; 0 , , <= 0 . By ^ n n n+1 n a well-known p r o p e r t y o f u : y(B) = i n f y(0 ) n which i s #-absolute. The r e s u l t f o l l o w s . The next theorem i s the main r e s u l t o f t h i s s e c t i o n . Theorem 2.24 L e t a code a B o r e l s e t , then ( y(B a)=0 ) i s #-absolute. Pro o f : Our s t r a t e g y i s r e m i n i s c e n t o f Lemma 2.5. (a) ( y ( B a ) = 0 ) i s p r e s e r v e d under the e x t e n s i o n IE\u2022 -\u00bb\u2022 IF . F o r 3 e to\" l e t 3 ^ ( 3 ) h o l d i f f ( 3 codes a 3 ^ - s e t ). From Lemmas 2.19(b) and 2.20(a), we i n -f e r t h a t ^ (3) i s (#-)absolute. From Lemma 2.23: ( ^ ( 3 ) & y(B ) ' = ' 0 ) i s #-absolute. Using Lemmas 2.4 and 2.19(c) we see t h a t i f ( y(B ) = 0 ) 33 e to W( ^ 6 ( B ) & B a c= B g & y(B ) = 0 ) holds i n IE , then by d e f i n i t i o n o f |=, i t h o lds i n IF . Thus Va e uit, : IE TR TP ) JE ( B a ). . 0 -> v w (B.^ ) = 0 , i f a i s a code. (b) ( y (JB^) = 0 ) i s p r e s e r v e d under the r e s t r i c t i o n IF -\u00bb- IE . For 3 \u00a3 to W l e t y ^ B ) h o l d i f f ( 6 codes a c l o s e d s e t ). Lemma 2.20 s a y s \/ ^ 3 ) i s (#-) ab-s o l u t e . Since \/(B) + ^ ( P \" ) * Lemma 2.23 i m p l i e s t h a t (\/(B) & V(Bg) = 0 ) i s #-absolute. I f ( V(B a) = 0 ) V0ea>u( (\/(B) & B B c B a) -*\u2022 y ( B B ) = 0 ) holds i n 3F then by d e f i n i t i o n o f f=, i t holds inJE . Thus Y a e ooW; y ^ ( B ^ ) = 0 y ^ (hf ) = 0 , i f a i s a code. The f o l l o w i n g lemma of Solovay i s a consequence o f the above theorem. C o r o l l a r y 2.25 (a) L e t B be a B o r e l s e t i n IE , then: ^JE ( B ) = yTP ( # B ) ' (b) L e t B be a B o r e l s e t i n JF r a t i o n a l over IE , then: ^IF ( B ) = ^IE ( B n ] R I E ) ' Pro o f : (a) Consider B as a measurable s e t ; by Lemma 2.22 B = G^N, where G i s and 62 N i s n u l l (. measure zero ) . Mote t h a t i n t h i s case N must be B o r e l a l s o . Mw (#B) = Mw (#G) - y-pdN). ( b y 2.19 ) = y E (G) - 0 (. by 2.23, 2.24 ) = y E (B) -(b) follov\/s from (a) and C o r o l l a r y 2.17 . The proof above might l e a d us to c o n j e c t u r e t h a t the r e s u l t h o l d s f o r B measurable. Our p r e s e n t development o f f e r s no ground f o r t h i s c l a i m , and the reason i t doesn't underscores the whole r a t i o n a l e o f t h i s s e c t i o n . Because the codes range over the s e t to w, we may express u n i v e r s a l 1 statements about B o r e l s e t s as IT^ formulas. Even i f we c o u l d apply codes to the measurable s e t s , t h e i r c a r d i n a l i t y 2*\u00b0 would be too l a r g e ( 2 ) f o r t h i s treatment. In such a case we have no i n f o r m a t i o n r e g a r d i n g a b s o l u t e n e s s , even f o r simple formulas. 3 63 Section 3 ; The Random Reals We assume the existence of a standard t r a n s i t i v e ground model M of ZFC which i s a submodel of IK countable i n IK . A l l our subsequent model constructions w i l l be b u i l t from IMC. Let IR denote the r e a l numbers of IK , and B denote the a-algebra of Borel sets of reals i n IMC. Let N denote the CT-ideal of Lebesgue measure zero sets i n IMC. D e f i n i t i o n 3.1 B* = IB \/N i s the quotient algebra of equivalent classes of Borel sets [A] such that B e [A] ( B = A (mod N ) ) i f f y (A A B) =0 ( A i s symmetric d i f f -erence ) . Proposition 3.2 B* i s a complete Boolean algebra s a t i s -fying the countable chain condition. Proof: See [8], p. 67 . This proof involves AC, but M |= AC, thus B* i s IMC-complete. IR __ = IR H IMC i s countable, so most reals f a l l outside IMC i t . A large portion of these extraneous reals have a spe-c i a l property which i s instrumental i n the construction of our generic extensions of IMC. D e f i n i t i o n 3.3 IR * i s the set of those reals belonging to no measure zero Borel set which i s r a t i o n a l oyer IE . For x e I * , we say that the r e a l number x i s random over JE . Since JM i s fixed we w i l l write IR * as IR *, and IMC c a l l the elements of IR * random r e a l s . While i t i s clear that there are no random reals i n IMC C y({x}) = 0 ), the existence of random reals i s immediate: Lemma 3.4 IR * i s a Borel set having measure zero comp-lement. Proof: As IMC i s countable we may enumerate the Borel sets of measure zero, r a t i o n a l over M. Their union i s a Borel set of measure zero and equals IR ^ JR * . Of course IR * i s not r a t i o n a l over IMC, and since Q \u00a3 JM, each random r e a l i s i r r a t i o n a l . Solovay ( [23], pp. 4, 33 ) remarks that the random reals are characterized by. '.random1 binary expansions: for large n, any block of 2n consecutive entries i n the expansion contain approxi-mately n zeros and n ones. D e f i n i t i o n 3.5 Let G be an JM-generic u l t r a f i l t e r on B*: x G = { r : r \u00a3 Q , [ (r,\u00b0\u00b0) ] \u00a3 G } . It i s easy to show that x p i s a (left) Dedekind cut, G and as such can be i d e n t i f i e d with the r e a l : sup x^ . x^ t e l l s us a great deal about the structure of G. We define a. complexity function A mapping the codes into the ordinals. D e f i n i t i o n 3.6 fa) If a code Y e toW s a t i s f i e s Y CO)- = 0 (mod 3), define ACy) = .0 . (b) If Y(0) = 1 (mod 3), l e t Y iCj) = y ( J ( i , j ) ) and define XCY) = supUCYi) + 1) \u2022 i Cc) If YCO) = 2 (mod 3), define A(y) = MB) + 1, where 6(n) = YCn+1) S t r i c t l y speaking, the proof below i s a t r a n s f i n i t e induction on A M = A f \\3MC, which maps (to ) M into 9 but we omit the extra notation. Lemma 3.7 Suppose B i s a Borel set i n IMC and G i s an M-generic u l t r a f i l t e r on B*, then: x G E #B i f f [B] e G . Proof: We set IE = IMC, IF = IK , and define # as i n the l a s t section. Let B = B^ where y e (toW) M . (a) Suppose y(0) = 0 (mod 3), then By = [ r y ( l ) ' r y ( 2 ) ] = # [ r Y ( D ' T y ( 2 ) ] = # B y ' x\u201e e #B -\u00ab-\u00bb\u2022 r . . < x < r . . -\u00ab-\u00bb\u2022 G y y CI) G Y ( 2 ) [ rY ( l ) ' to) \u00a3 G & ( r Y ( 2 ) ' \u00b0\u00b0) 1 G ^ [ B Y ] \u00a3 G (b) Suppose y(0) = 1 ( m o d 3) ' then B = UB Y i Y i and #B = y#B by Lemma 2.19. By induction Y j_ Yj_ hypothesis and f i l t e r properties; xr e #BV . xr e \u2022 U #B *+ a i t x r \u00a3 #B . \\ s* T G i Y i : \u00ab Ti 3i ( [B J \u00a3 G ) ( as A (Yi) < A (y) ) r i \u2022 . [B ] \u00a3 G C as l[Ey f' = [B J ) i 1 (c) Suppose y(0) = 2 (mod 3), then B v = 3R B 0 , where 3 (n) = y(n+l) . Y p xn \u00a3 #B, ^ x . i #BQ ( by Lemma 2.19 ) G Y G p -H. [B g] \/ G ( as X(3) < A(y) ) -<-*- = L B y] e G ( as G i s an u l t r a -f i l t e r ) . The r e s u l t follows by t r a n s f i n i t e induction on complexity of codes. ( An argument simi l a r to that of Lemma 2.9 proves that every code has a complexity. ) We can use Lemma 3.7 to estab l i s h a natural b i j e c -t i o n between the generic u l t r a f i l t e r s and the random reals Lemma 3.8 x \u00a3 3R* i f f x = x. for some generic u l t r a f i l -\u2014\u2014 ter G . Proof: (a) Let x e M* . Define G = {[B] : x e #B} Let A,B e JB ; i f A = B (mod N) then x e #A -\u00ab->\u2022 x e #B R, as y ( A A B ) = 0 and y (#A A #B) = 0 ( Lemma 2.19 and Theorem 2.24 ) Thus: IBJ \u00a3 G -\u00ab-> YA e IB], x e #A . . . . (1) By. Theorem 2.24: [01 \u00a3 Gv . . ... (2) If [A], [B] e G x then x e #A, x e #B, and x e #A n.#B = #(. A n.B ) by Lemma 2.19 . Thus: [Ah B] = [A] \u2022 [B] e G . ... (3) Let [A] \u00a3 G and [A] < [B], then by (1): VC e [B] , x e C and so [B] e G . (4) (2) \u2014 (4) imply that G i s a proper f i l t e r . X G i s obviously maximal, and i s thus an u l t r a -x J ' f i l t e r . Let S B*, S e M , IS e G . Since B* obeys X the countable chain condition, there i s a M> countable c o l l e c t i o n of Borel sets: { A , ... ,A , ... } s M such that [A ] \u00a3 S, for each i \u00a3 co, and: I[A ] = ES h. Y . x Y r ( see [8], p. 61 ). Let y (0) = 1 (mod 3), y ( J ( i , j ) ) = Y \u00b1 ( j ) then A = u A and [A ] ' i ^ i ' \u00a3 G . Hence x \u00a3 #A by Lemma 3.7 . Lemma x 2.19 implies #A = U# A , so Hi ( x \u00a3 #Ay ) i i i and [A ] \u00a3 G H S ( Lemma 3.7 ). This estab-yL x lis h e s the genericity of G \u201e. Note that even X though |M| = , the countable chain con-d i t i o n i s required. 68 Lemma 3.7 now gives: x = x_. , where G = G G x (b) Conversely, l e t x = x_, for some generic G u l t r a f i l t e r G on B*. For each Borel set B Y r a t i o n a l over M and s a t i s f y i n g JJ (B ) = 0, Theo-rem 2.24 implies i i M ( B \u2122 ) = 0 , and so [B^] \u00a3 G. It follows from Lemma 3.7 that x \u00a3 B \u2022 Y Corollary 3.9 For each x e l * , G = { [B] : x e #B } \u2014\u2014\u2022\u2014~~~~\u2014\u2014\u2014\u2014 \u2014\u2014\u2014 i s an M-generic u l t r a f i l t e r on B*. These res u l t s give us some notation and terminology: (a) Each x e 3R * has an associated generic u l t r a f i l -ter G on B*. x (b) Each generic u l t r a f i l t e r G on B* has an associated random r e a l X\u201e . De f i n i t i o n 3.10 For x e 3K , l e t M fx] be the least tran-s i t i v e submodel of IK extending M and containing {x} , i f t h i s exists; M[x] = IK otherwise. We w i l l only use the above notation where i t i s well defined, primarily by the r e s u l t below. Lemma 3.11 For every x e IR*, M [x] = M [G ] . Proof: M [ G ] i s the least t r a n s i t i v e submodel of IK x extending M and containing {G1} ( see proof [9] , p. 56 v i a absoluteness^ or Lemma 4.8 for d i r e c t proof of a stronger r e s u l t )_. e IMCJG .] , by De f i n i t i o n 3.5 .. If I ? M i s a t r a n s i t i v e submodel of IK ( hence IN i s standard ) , and x e IN , then G e IN by Corollary 3.9, and so M[G ] c ]N , i . e . X x e \u201e M [ G ] <= IN IMC D e f i n i t i o n 3.12 For a given generic u l t r a f i l t e r G on B*, B* define X\u00a3e IMC as: dom(xG). = { r : r e Q, [(r,~)]eG; } ^ ( r ) = [(r,\u00b0\u00b0)] B* X~ i s c a l l e d the canonical random r e a l i n M x^ , \u2014(j \u2014(j names x G , i . e . i G (x^J = x G . For G understood, x = x^ , . We r e c a l l t h i s restatement of Lemma 3.8 of Section 0. Lemma 3.13 For each formula (y) i f f I (y,P~) r P e IMC. This gives: IT M [ y ] h *(Yf?) i f f II *(x,f) ] ] e G y , y e I R * , p e I M C . Theorem 3.14 For each formula cj), the set E = { y e IR : IMC [y] ( = (y) } . 70. By Lemma. 3,4, E' = E (modN ). Then by Lemma 3.13: y e E' <-> l t y ( x ) J eG^ . Let y e \u2022\u00ab\u00bb-. code the Borel B such ' JM Y that: [B^ .] = I ty (x) I . Then for a l l y e JR*: y \u00a3 E' [B ] \u00a3 G y \u00a3 #B Y Y Y Therefore: E' = #B^ (modN ) , E = #B^ (modN ) , and E i s Lebesgue measurable. The above theorem e a s i l y generalizes by adding para-meters i n JM, and i t i s th i s form of Theorem 3.14 that finds application i n Section 5., 71 Section 4 : The Levy Algebra Every Boolean algebra i s a p a r t i a l l y ordered set, but seldom does a p a r t i a l l y ordered set have the necessary struc-ture to make i t a Boolean algebra, much less a complete Boolean algebra. Fortunately, a standard technique exists which transforms any given p a r t i a l l y ordered set into a complete Boolean algebra. If P i s a p a r t i a l l y ordered set we write RO(P) for the regular open algebra of P. This i s obtained by imposing the order topology on P ( with basic open sets [p] = { q : q <_ p } ). The elements of RO(P) are those open sets U which are regular ( i . e . U = U\u00b0 ). RO(P) i s a complete Boolean algebra. Complete d e t a i l s are to be found in [8] ( p. 25 ) and\/or [25] ( pp. 1 4 - 1 7 ). As usual, c f ( a ) denotes the c o f i n a l i t y of an ordinal a . D e f i n i t i o n 4.1 Suppose M |= K i s a cardinal & cf (K) = to. P i s defined by: M (= p e P \u00ab-\u00bb- ( 3n e to ) ( p:n-H< ) , and i s p a r t i a l l y ordered by: P ^ q ^ q ^ P -For each cardinal K, P i s simply a c o l l e c t i o n of f i -nite functions i n M with range i n K. The ordering of P i s reverse of the usual inclusion ordering. The p a r t i a l l y ordered set P gives us a complete Boo-K \u2022 lean algebra L =\u2022 RO(P ), c a l l e d a Collapsing algebra K K 1 C the reason for t h i s name: for any generic u l t r a f i l t e r G 72 on L f the function UG maps oo onto K, and so K \"collap-IS ses\" onto oo and i s countable i n M [G] . See [23] , p. 8 ) . For the present we s h a l l assume there i s a (_ strongly ) inaccessible \"cardinal X. The ramifications of t h i s assump-tio n are discussed i n the next section. The family { P : cf (K) = oo, K < A } forms a normal l i m i t i n g system ( see [25], p. 193 ). It follows that the associated family { L : cf (K) =oo, K < A } i s an example of a d i r e c t system of complete Boolean algebras. An exhaustive development of t h i s topic i s found i n [25], pp. 183 - 195 . De f i n i t i o n 4.2 A Boolean algebra B s a t i s f i e s the K-chain condition i f each p a r t i t i o n of unity i n B has c a r d i n a l i t y less than K. In the case where K = oo, we say that B s a t i s f i e s the countable chain condition. Two members a, b of a Boolean algebra ( or p a r t i a l l y ordered set ) B are said to be compatible i f there exists a nonzero c e B such that c _< a and c _< b, otherwise they are said to be incompatible. Since a p a r t i t i o n of unity i s a maximal family of pairwise incompatible elements of B, the K-chain condition implies that no family of p a i r -wise incompatible elements of B has c a r d i n a l i t y K, or greater. A p a r t i a l l y ordered set s a t i s f i e s the K-chain condition i f i t contains no s t r i c t l y descending chain ( to-t a l l y ordered set ) of c a r d i n a l i t y K . The proposition below gathers together a number of technical r e s u l t s , mainly from the above reference, which support the work of t h i s section. We quote them without t h e i r lengthy but straightforward proofs, some of which derive from the work of Engelking and Karlowicz [5]. Proposition 4 . 3 (a) For each K < A such that cf (K) = co, P = ,UP K a<\\ a (b) P = UP s a t i s f i e s the. A-chain K x y \u00a3 G1.. Complementation and the existence of 0 must not be taken for granted i n P; that i s why we are forced to simplify the notion of f i l t e r from the o r i g i n a l Boolean algebraic case. D e f i n i t i o n 4.6 G' i s an M-generic f i l t e r on P i f f for each dense set S c P, S \u00a3 M, we have S f\\G1 ? 0 . G' c P above i s also c a l l e d a generic set of forcing conditions i n the l i t e r a t u r e . We w i l l not delve into the method of inducing a gen-e r i c f i l t e r GV on P, given a generic u l t r a f i l t e r G on L, or that of inducing G from G1 on the other hand. The l i t -erature contains ample treatment of t h i s (. see [25] , pp. 25 - 32, es p e c i a l l y p. 30; see also [9], pp. 48 - 52 )\u201eand we s h a l l never need to appeal to the mechanics of i t . Suffice i t to say that RO induces a one-to-one correspon-dence between the generic u l t r a f i l t e r s of L and the generic f i l t e r s of P. Why have we defined L, and what' properties does i t have that simpler, more fa m i l i a r algebras do not? We have already mentioned that by i t s d e f i n i t i o n , L s a t i s f i e s the A-chain condition, and that t h i s fact i s very useful i n both t h i s section and the next. L has however, a very strong and unusual property having c r i t i c a l impact on Solo-vay 's application of random reals to the measure problem. It i s to t h i s property of homogeneity that the duration of t h i s section i s devoted. A l l of the p r i o r r e s u l t s we have c i t e d are e a s i l y ac-cessible i n the l i t e r a t u r e , and so we have quoted them with-out proof. The proof that L i s homogeneous i s not well re-presented elsewhere, so i t deserves a detailed treatment here. Suppose A i s a complete subalgebra of L, and g i s an automorphism on A. We say that g l i f t s from A to L i f there i s an automorphism g 1 of L whose r e s t r i c t i o n to A i s g. 77 g 1 i s c a l l e d an extension of g. I t i s by no means clear what conditions we might impose on A to ensure the l i f t i n g of each g e Aut CA) . Letting Aut(A) denote the set of automorphisms on A, we say that a complete subalgebra A of L has the l i f t i n g property i f each g e Aut(A) l i f t s . D e f i n i t i o n 4.7 L i s homogenous i f each complete sub-algebra A of L s a t i s f y i n g |A| < X has the l i f t i n g property. The term \"homogeneous\" has various meanings i n the l i t e r a t u r e . For our purposes, the strong notion of homo-geneity we use i s necessary. Let us f i r s t review some e a s i l y obtainable information. L i s complete, as i t i s a regular open algebra. A Hahn-Banach type extension argument can be employed to show that complete Boolean algebras are i n j e c t i v e - ( see [8], pp..132 -143 ), i . e . they s a t i s f y the commutative diagram below for any Boolean algebras A and B: B where e i s any monomorphism, and h i s any homomorphism. Having fixed a l l of the above p a r t i c u l a r s , i n j e c t i v i t y 78 simply means that a homomorphism f exists which completes the diagram. One of the main consequences of i n j e c t i v i t y i s the fact that homomorphisms on subalgebras into L extend to homomorphisms on L. This follows d i r e c t l y from the d i a -gram by l e t t i n g B = L, and e be the incl u s i o n map. We know then, that an automorphism on a subalgebra A of L extends to some homomorphism on L. Using a Hahn-Banach type argument, we can show that embeddings ( complete mono-morphisms ) of subalgebras extend to embeddings of L. Un-i v e r s a l techniques show us then, that automorphisms on any subalgebra extend to embeddings of L into L. To show that L i s homogeneous however, we must use properties s p e c i f i c to L. The path we w i l l take involves a novel use of Boolean-valued techniques. Though we are confronted with a non-l o g i c a l problem concerning L, we w i l l f i n d that much alge-braic information about L i s r e f l e c t e d i n ]MCL. We r e c a l l that DMC i s our t r a n s i t i v e ground model of ZFC. Lemma 4.8 Let A be a subalgebra of L, and h be an auto-morphism on A. In 3MCL there i s an u l t r a f i l t e r G, on A such that for each a \u00a3 A: h h(a) = [[ a e G, B . n Proof: G i s the canonical u l t r a f i l t e r on M L. Since G(l) = A C D =1, we have: M h AH.G f p. and the Maximal P r i n c i p l e (\/Lemma 0.26 )_ de-fines G such that: 3MCL (= G a = AflG . XT. Likewise, we use the Maximal P r i n c i p l e to de-fine G, on A: h M L \\= a e G, h(a) v e G, . h A An elementary c a l c u l a t i o n y i e l d s : I a E G A 1 = a , for each a e A . So we have: I a e G h ] = I h(a) v e G A B = h(a) . From t h i s , and the fact that h i s a monomorph-ism, simple calculations give: (a) I 0 \u00a3 G h JJ = 1 , (b) a' <_ b implies [ a e ] < I b \u00a3 G h 1 , (c) I (a-b) v e G h 1 = [[ a e G h & b e G h JJ , (d) \\ (-a) v e G h J J = - | I a \u00a3 G h J ] . We conclude: L i ^ IMC h G, i s an u l t r a f i l t e r on A. \u201e h We refer to G, as the u l t r a f i l t e r on A associated with h h. This lemma i s understated i n the sense that h could just as well have been an embedding of A into L. Even so, we have not extracted a l l the information about G^ that i s re f l e c t e d in h. Corollary 4 . 9 IMCL f= G h i s (IP(A) ) v-complete. 80 Proof: h i s complete. The main work within L i s car r i e d out by the follow-ing r e s u l t , which i s our modified version of a theorem due to Jensen ( see [9], pp. 7 5 - 7 6 ). To prepare for i t , we mention the following items that are necessary i n the proof: (a) A subalgebra of a Boolean algebra i s said to be regular i f suprema ( and infima ) of subsets common to both algebras correspond to the same value i n each algebra. It i s e a s i l y v e r i f i e d that complete subal-gebras of complete Boolean algebras are regular. (b) The f a c t that A = [* ] , which i s proved i n the next section ( Corollary 5.2 ) using no information dependent on Lemma 4.10 . Lemma 4.10 Let A, B be complete subalgebras of L having c a r d i n a l i t y less than A , and l e t A be a com-plete subalgebra of B. Each automorphism on A l i f t s to be an automorphism on B. Proof: Let h be an automorphism on A, and be i t s associated u l t r a f i l t e r on A. Using the Max-imal P r i n c i p l e again, we induce a f i l t e r G* on B: M L \\= (YxeB) ( xeG* ++ (3y \u00a3A) (y G* . A mapping g may now be defined for each b e B: g(b) = I b E G' 1 . The following Boolean suprema, evaluated i n th e i r respective algebras, are equal: [[ b e G' I ( B ) = II b e G' ]] ( L ) , as B i s a regular subalgebra of L. Hence g maps B i n t c B . 82 Using the fact that G' i s an u l t r a f i l t e r and that i s i n f e c t i v e , we have for each a, b e B: g(-a) = t[ (-a) - e G' J = I a \u00a3 G' JJ = - 1 a e G'.'l = -g(a) , g(a\u00abb) = [[ (a\u00abb) v e G1 ]] = [[ a e G1 & b e G1 ]] = g (a) \u00bbg (b) . Thus g i s a homomorphism. Si m i l a r l y , we can use (JP (B) ) \"-completeness of G' i n M L to show that g i s a complete homomorphism. For each a e A, we have: g(a) = [ [ a \u00a3 G I J = [[ a e ] = h(a) , so g i s an extension of h. There are many u l t r a f i l t e r s G1 for which the above calculations hold. We w i l l select one of these which ensures the i n j e c t i v i t y of g. By Lemma 4.4(b), there i s a pairwise incompa-t i b l e family { a^ \u00a3 L : b \u00a3 B, b ^ O } such a, *c 4 0 for each a, , and each c \u00a3 A where c 4 0. b b By Lemma 0.25, there exists t e M such that a^ <_ [t t = b D \/ for each b e B where b 0. We pick t 1 \u00a3 3MCL such that: I ( t 1 \u00a3 B) & ( t 1 = -t) I = 1 , i . e . t 1 i s the complement of t i n B. Now we define G' as before, but with the added proviso: [ II t' ft G* I < I t \u00a3 G' I . This amounts to generating G' from E = G*U{t} i f E i s pairwise compatible, i . e . 83 M L h ( G' i s a (IP (B)) v-complete u l t r a f i l t e r on B ) & ( G* c= G' ) & ( t' \u00a3 G* -> t e G' ) . Let b e B, and b ^ 0 . g(b) = lb \u00a3 G'l > lb = tUvttt e GM >. lib = t l - I f \u00a3 G*J >_ Vo = tj'[[-(b) \u00a3 G*IJ = Eb = tU \u2022 I (-b)v \u00a3 G*] a b \u2022 -((-b)*) . Since -b < 1; (-b) * < 1, - ( ( - b ) * ) ^ 0, and so g(b) \u2022\u00a3 0, by d e f i n i t i o n of a^. Thus ker(g) = 0 and g i s i n j e c t i v e . Since B i s i n j e c t i v e , the diagram indicates that each monomorphism on B ( such as g ) has a r e t r a c t i o n f, i . e . an epimorphism f such that f\u00b0g i s the i d e n t i t y map on B. If f i s a re-tr a c t i o n for g, then f i s complete, and the kernel of f i s a complete Boolean i d e a l . Id-eals of th i s type must necessarily be p r i n c i -pal , i . e . there i s an element u e B such that ker(f) = [u] = { v : v j< u } . If rng(g) ^ B, then u ^ 0 and u \u00a3 rng(g) . Define: E = { t \u00a3 B : [ [ t \u00a3 G ' ] ] ^ u } . Since dom(E) = { t : t \u00a3 E } , and E ( t ) =1 for each t \u00a3 E , a simple c a l c u l a t i o n y i e l d s : V V [ E ] i s a fixed point for any g e Aut(L). Hence [[cj)]] i s either 0 or 1. The same reasoning gives us a 0-1 law ( see [19], p. 408, and [25], p. 171 ) for formulas whose variables range over JM. Corollary 4.14 Let cj) be a formula with n free variables For each x,, ... ,x e M : 1 n [[cj)(x1, ... fx n)I e {0,1} . Proof: For each b e L^{0,l}, we may use Theorem 4.11 to construct g, e Aut(L) such that 88 g^(b) f b . For example, define the automorphism e on the subalgebra {0,b,-b,l} by e(b) = -b , and l i f t e to g^ on L. Let b = I cb ( X l, ... ,x U . If b \u00a3 {0,l}, Corollary 4.13 implies: V V b = [[0 (g*x l f . . . ,g\u00a3x n ) I = gbf[<{> 0 ^ , ... \/X n)I ^ b . Hence b e {o,l} . It i s possible to prove Theorem 0.22 from the above, since bounded formulas are of t h i s type, and t h e i r Boolean values are non-zero when they are v a l i d i n M. In the next section, we define L t as the complete sub-algebra of L generated by rng(t), where t e 3MC L. Lemma 4.15 If |rng(t)| < A, then |Lfc| < A . Proof: Since P corresponds to the basic open sets i n the topology of L, P i s dense i n L. Using the A-chain condition, we can thus fi n d for each a e L, a subset S cr P such that |S I < A, and a 1 a 1 a = ES . Let S = U{ S : a e rng(t) } . Since | S^ | < A, there i s a K < A such that S cr P . L = RO(P ) and S c L , so L c L , and K K K t K l L t l 1 l L J <- X ' 89 A generalized form of Corollary 4.14 now follows, Theorem 4.16 Let t e M be such that dom(t) c { x e M \u2022 } , and | rng (t). | < X . Then; x K (t) U - Since u = b yi e l d s a contradiction, we conclude that b \u00a3 L, . A l l of the constructions i n t h i s section were carried out within M, and so our many uses of AC i n various forms have been proper. In spite of the forbidding number of technical results i n t h i s section, only two of them have any application i n the sequel. These are: Proposition 4.3 (c), used i n some c a r d i n a l i t y calculations, and Theorem 4.16 90. above, which i s our s o l e a p p l i c a t i o n of the homogeneity 91 Section 5 : The Model Of Levy In previous sections we have imposed a number of plau-s i b l e r e s t r i c t i o n s on the ground model ZMC. To begin t h i s section, we add one more r e s t r i c t i o n to the l i s t : namely that M s a t i s f y the following axiom. Axiom I There exists an inaccessible cardinal. The above statement i s much stronger than the axiom A we used i n section 0 to j u s t i f y the existence of IK . Tar-ski [26] shows that a model of ZFC can be constructed i f there exists an inaccessible cardinal ( see also [14], pp. 159-63; [4], p. 109-10 ). Similar arguments show that such a model may not be a model of I ( e.g. [4], p. 110; [9], p. 37 ). Hence model existence axioms such as A cannot r imply I. The assumption M |= I i s , i n d i r e c t l y , an added con-s t r a i n t on our assumptions regarding IK . Though axiom A i n i t s present form i s not s u f f i c i e n t to provide such a IK , we s h a l l bypass t h i s problem for the present. Taking M to be the ground model containing an inac-cessible X , we may construct within M the Levy algebra L of Section 4. We f i x an M-generic u l t r a f i l t e r G on L, and i n the course of t h i s section construct the following 'tower' of generic extensions: 92 Since M \\= AC, each generic extension also s a t i s f i e s AC, and therefore ~| LM as w e l l . However, the r e s u l t s of the l a s t section w i l l provide that a large family of sets of r e a l s i n IMC [G] are Lebesgue measurable. To begin, we check the behaviour of cardinals with re-spect to the above tower. Lemma 5.1 Let IE be. a model of ZFC, B be a Boolean alge-bra i n IE , s a t i s f y i n g the countable chain con-d i t i o n i n IE , and l e t H be an IE -generic u l t r a -f i l t e r on B. Then: $ = ( K i s a cardinal ) i s absolute between IE and IE [H] . Proof: F i n i t e cardinals are absolute, so we assume K _> . Suppose IE =^ $ , and. IE [H] |= ~~| $ .. Then there e x i s t s a Boolean-valued function f e IE such that for 6 < K b = [\u2022 dom(f) =6 & rng(f) = < Tj ? 0 . Since we have not defined the Boolean-valued notions of dom(f), or r n g ( f ) , we w i l l assume f s a t i s f i e s D e f i n i t i o n 0.29 and define: 93 b = n Z I. (g,y) B e f J \u2022 H , .E tt.(a,Y)B e f- H a<6 ' Y < K y < K a<6 ft 0 . . V \" B We l e t b ( a , 3 ) = b \u2022 [[(a, 3) e f J '. Two facts emerge: (a) From D e f i n i t i o n 0.29(c) and Theorem 0.22, v v 6 fi y\u2022 + b ( a , B ) *b (a,Y) < b \u2022 EB = YI = 0 , as f i s a Boolean-valued function. (b) Since b ft 0, V3 < < , 3 Y > 3 , 3 a ( b ( a , Y ) f^ 0 ) . From (b) we have: |{ 3 <-K : 3 a , b ( a , 3 ) 0 }| = K . Since 6 < K, 3 a 0 such that: | { 3 < K : b ( a Q , 3 ) ^ 0 } | = K . By (a), { b ( a D , 3 ) : 3 < < } i s a set of p a i r -wise incompatible elements of B having cardin-a l i t y K . This v i o l a t e s the countable chain condition. $ i s therefore preserved under the extension IE -*\u2022 IE [H] . A routine argument shows that $ i s preserved under r e s t r i c t i o n . The c o r o l l a r y below r e c a l l s some terminology from Sec-tio n 4. P i s the c o l l e c t i o n of f i n i t e sets of t r i p l e s p = \u2022 { (a i,n i, B i n i < k s a t i s f y i n g : (a) n. eco, 3- < a. < A I I l (b) ( a,n , 3 0 ) \/ . C a,n , 3 1 ) e p 3 Q = 3]_ \u2022 P i s p a r t i a l l y ordered by p. < q i f f p => q. L - RO(P) . I f G is. an JMC-generic u l t r a f i l t e r on L, l e t G' be i t s i n -duced JMC-generic f i l t e r on P. We t h i n k of p e P as a func-t i o n having f i n i t e domain <= ^ .xoo, wit h p.[ (a^,n^)] = 3j_ < ou. C o r o l l a r y 5.2 JM [G] Proof: Le t 6 = to and K = A i n the proof o f the preceeding lemma. Using the f a c t t h a t L obeys the A-chain c o n d i t i o n , we i n f e r JMC r G1 ^: > (^^ . For each a < A, we de-f i n e : f a = { (n,3) : { (a,n,3) }\" e G'} . From (b) above, each f i s a f u n c t i o n a i n M [G] , and f : to ->\u2022 a . For a \/ 0, Yn \u00a3 to, n e dom(f a) : Let A Q = { h e P : (a,n) e dom(h) } . Since a > 0, A D i s dense i n P. By gen-e r i c i t y t here i s an h \u00a3 G ' n A Q such t h a t 33, h[(a,n)] = 3 . Then {(a,n,8)} e G ' , as G ' i s a f i l t e r . n \u00a3 d o m ( f a ) . Y3 < a, 3 e r n g ( f ): Let A 1 = { h \u00a3 P : 3n \u00a3 to, h[(a,n)] = 3 } A^ i s dense i n P. The r e f o r e t h e r e i s an h e G 1 r\\A 1 such t h a t h[(a,n)] = 8, so {(a,n,8)} e G 1 and 3 e rngCf^) . We conclude t h a t f o r a l l a < A, f maps to onto a. Thus A <_ ) JMC [G] 95 . Def i n i t i o n 5,3 Let s e M.[G] , s \u00ab= IMC, and s e M L be a name for s. L i s the complete sub-algebra of L generated by rng (s_) . Lemma 5.4 For each x e M, and s e IMC [GJ , Ix e sj e L g , where s_ names s. Proof: Let A be a complete subalgebra of L containing A v {0 rng (s_) . For each x e M , x e I M C a s x e M ' and {0,1} i s a subalgebra of A. Since dom (s_) L ^ c M s, [[z = x] e A for each z e dom (s_) . Thus: Ix e s] = E s(z)-IIz = x I ] \u00a3 A , z e dom (s_) by completeness of A. Our attention turns now to the f i r s t extension of the tower. The role of t h i s extension concerns the d e f i n a b i l -i t y of those sets of reals i n IMC [G] we wish to be Lebesgue measurable. We say that a set E i s definable ( i n IMC [G] ) from s \u00a3 IMC [G] , i f there i s a formula having free v a r i a -bles only x, s, such that E = { x : IMC [G] |= cf>(x,s) } . For the rest of t h i s section, our in t e r e s t w i l l focus on those sets of reals i n IMC [G] definable from a sequence of ordinals. Thus s e M[G] i s a function with domain co, ranging over , the ordinals of IMC ( IMC and IMC [GJ have the same ordinals; see [25], p. 128 ) . In Section 3 we found a connection between the generic u l t r a f i l t e r s over B* and the random r e a l s . The next r e s u l t gives us a general connection between the generic u l t r a -f i l t e r s induced by G on L and the extensions ML Isj . Lemma 5.5 Let s e name s e M[G] . Then: 1 * 1 . 8 1 = M [G A j . ] Proof t Since G i s an M-generic u l t r a f i l t e r on L, G C\\ L i n h e r i t s a l l the properties necessary' to make i t an M-generic u l t r a f i l t e r on L . M [ G n L A ] i s therefore a true generic exten-sion. Since for a l l x e M : M [G] [= x e s i f f I x e s I e G i f f [ x \u00a3 s ] e G Pi L, i f f we haves Let U s I f , and s e W . For each a e 0, defines (s rb) ( 3 x e dom (s) ) ( b = s (x) & b e G ) o \u2014 \u2014 \u00b0 (b) v <|>* (b) ) ] . These formulas re f e r to the following sets: A G = rng(s) , A = f-c : c e U AD} ; for a even, A a =\u2022 {EF : r <= U^,} ; for a odd. Let G a = GQA a . Since L e IMC c UN , and G D cr L, the separation axiom implies G 0 E l . If a > 0 i s even, G a = { b : \u00b0 (b) }, and i f a i s odd, G a = |\u2022 b : ^ \"(b) } . So i f G R e IN for each 6 < a, then G e IN . a We conclude that GHL = { b : ^(s,s_,b) } U G e IN , and that M [GHL ] i s the <\u2022 - a -least model of ZFC extending IMC and containing {s}. ->-We say that b = b(p) e IE i s uniformly JE -definable i n p i f there i s a formula \u00a7 with free variables among p, x, ->- -\u00a5 such that for any set of values p Q of the parameters p: b(p 0) = { x : IE \\= (J> (p 0 ,x) } . It i s usually not convenient to mention a l l the parameters i n p, p a r t i c u l a r l y those which are always fixed. In the co r o l l a r y below, we make no reference to such parameters as L or G for t h i s reason. Where no parameters are needed, 98 we w i l l drop the a,dverb \" u n i f o r m l y \" . C o r o l l a r y 5.6 GO.I' i s u n i f o r m l y M [s] - d e f i n a b l e i n g s, s\u00bb Our f i r s t diagram l e f t out an unusual a r c h i t e c t u r a l de-t a i l o f tower. The second e x t e n s i o n i s r e a l l y a s i m u l t a n -eous g e n e r i c e x t e n s i o n , r e v e a l e d below: As we s h a l l see, the upper e x t e n s i o n apparatus i s a a n a t u r a l r e p e t i t i o n of the f i r s t e x t e n s i o n , u s i n g the r e a l nunber y i n the p l a c e o f s. The lower e x t e n s i o n uses the r e s u l t s of S e c t i o n 3, on the h y p o t h e s i s t h a t y i s \"almost always\" a random r e a l . Roughly speaking, d e f i n a b i l i t y a s-pec t s are handled by the upper e x t e n s i o n and measure t h e -o r e t i c a spects are handled by the lower e x t e n s i o n . How do we know t h a t these e x t e n s i o n s agree? Lemmas 3.7 and 5.5 ensure t h i s . The s p e c i f i c p r o p e r t i e s o f s may now be used t o our advantage. Lemma 5,7 I f s i s a countable sequence of ordinals i n JMC [G] , i t has a name\" s_ e JMCL such that; |rng(s)| < X . Proof: By Lemma 0.40, we pick s_ so that to v .. I s_ e C. Q M flu) 1 e G, where u i s some set in JM. For each n e oo, K = | { a : I (n ,ap e s ] ] ^ 0 } | < X , since L obeys the X-chain condition. By re-gu l a r i t y of X we have: | rng (s) | <_ lim ic < X . Corollary 5.8 With s_ as above, | L | < X . Proof: By Lemma 4.15 . The next lemma i s the culmination of a l l our work on the Levy algebra L and i t s associated model M [G]. The existence of X and the r e s u l t i n g homogeneity of L are cru-c i a l factors i n i t s proof. The lemma introduces a reduc-tio n i n the d e f i n a b i l i t y of sets of reals i n M[G] that places them within reach of the upper second extension of the tower. Lemma 5.9 Let E be a set of reals i n M[G] which i s de-finable from a countable sequence of ordinals s e M[G]. E i s uniformly JM [s] [y]-definable in s, s, for each y e E. 100 Proof: We represent y e E by i t s Dedekind cut. There i s a formula whose only free variables are s, y, such that the following are equivalent: (a) y e E , (b) M [ G ] (= ' ( Y \/ Y \/ S f S ) . We have used the upper second extension to e f f e c t a reduction i n the c r i t e r i o n of membership i n E, from IMCfG] to M i s ] ly] . We w i l l apply the tools of Section 3 to t h i s 101 reduced c r i t e r i o n by way of the lower extension and obtain the f i r s t main theorem of Solovay. Lemma 5.9. renders E i n the parametric form of Theorem 3.14 .. The only fact we need check i s whether or not M[s] i s an appropriate ground model from the standpoint of Sec-tion 3. 3MC[s] can, i n fact, be shown to be countable ( by induction on rank, as i n [22], p. 36 1 ) . Instead, we w i l l use a more s p e c i f i c argument about cardinals that re-estab-lish e s Lemma 3..4 for the lower second extension. Theorem 5.10 Let E be a set of reals i n M[G] which i s M[G]-definable from a countable se-quence of ordinals s e JM [G]. E i s Lebesgue measurable. Proof: Each subset t of to i n M[s] has a name t e M \u00a3 with dom(t) = { h : n e ca } , and so determines a function f ^ : u ) -\u00bb- L t s_ defined by: V f (n) = II n e t I ( see 5.4 ) . Note that f. e M, . f^ = f -\u00bb-t t u Vn: II n e t J = I n e u 1 \u2022 \u2014 * I t = u 1 = 1 \u2014> M [s] (= t = u . Thus the number of subsets of co i n M[s] cannot exceed the number of functions i n M from to into L . Using the inac-s_ c e s s i b i l i t y of X i n M and Corollary 5.2: 102 2 * 0 ) M [ s ] < A = ( ) IMC [G] The c a r d i n a l i t y of the family of Borel codes i n IMC [si i s countable i n M [G] , so there can only be countably many Borel sets of measure zero i n IMC [G] which are r a t i o n a l over M [ s ] . Lemma 3.4 and Theorem 3.14 y i e l d the r e s u l t . 103 Section 6 ; The Model of McAloon We cannot expect any generic extension of a ground mo-del of ZFC to be a model of ZF + LM. However, the Levy model i s an example of a generic extension which comes very close to s a t i s f y i n g LM, i n that a certain large family of sets of reals are Lebesgue measurable. This fact suggests a new approach. Can we f i n d a suitable submodel of M[G] whose sets of reals f a l l within the above family? In t h i s section we follow the McAloon-Solovay construction of one such i n t e r n a l model 1 c M[G]. A l l the work of t h i s sec-t i o n i s c a r r i e d out within M [G]. If ]N |= LM, the problems of Section 1 regarding the needs of analysis and measure theory become pertinent. The model 3N which we construct w i l l s a t i s f y the axiom below, known as the P r i n c i p l e of Dependent Choices: DC : If R i s a binary r e l a t i o n on a nonempty set A such that for every x e A there exists y e A so that xRy, then there i s a sequence {x^} of elements i n A s a t i s f y i n g : ( Vn e to ) ( x Rx , ) . n n+1 It i s e a s i l y shown that DC implies AC^ ( see [1.0] , p. 23 )_, and so the r e a l analysis of ET i s \"normal\". It i s also true that AC implies DC ( see proof of Theorem 6.12 I. As an i n t e r e s t i n g aside, our construction of UN 1Q4 w i l l establish; two independence re s u l t s previously shown by other methods: CD; The Axiom of Choice i s independent of the other axioms of ZF ( Cohen, [2] ) . C2) AC i s independent of DC (. Feferman, [6] ). D e f i n a b i l i t y i s the central concept i n our construc-ti o n of 3N . Our ultimate i n t e r e s t i s with a family of sets which are the values of abstraction terms having spe-c i a l parameters. These notions are subject to hidden d i f -f i c u l t i e s of which mention i s now made. We f i r s t look at the simplest type of d e f i n a b i l i t y . A set x i s definable without parameters ( we write: Dwp(x) ) i f : x = { y : i s some formula with one free variable. For the class of such sets we write: DWP. Our previous uses of d e f i n a b i l i t y have.been informal i n the sense that no formula cf>0 of ZF has been exhibited for which: 0 (x) -\u00ab->\u2022 Dwp (x) . In general, the following version of Richard's paradox prevents t h i s . Proposition 6 .1 ~| Dwp (DWP) . Proof: Since DWP i s countable for the f i r s t or-der language of set theory, undefinable ordinals e x i s t . If (x) has only x free, note that y = { a : Y3 < a, x = a ) . (b) Dwp (DWP^ ) . 3E Proof: (a) Let a e DWP . Then there i s a formula ty such that a = {y : J\u00a3 \\= ty (y) }. Define tyQ(y,x) ( y e x -\u00ab-\u00bb- ty(y) ) . Then HE f= Yx ( Yy (tya (y ,x) ) <-+ x = a ) . Conversely, suppose there i s a formula ty (x) , with only x free, and 3E \\= Yx(iMx) x = a) . Then a = {y : 3E |= Yx ty0 (y,*\u2022)}, where: o ( Y f X ) ( y e x +-+ ty ix) ) . (b) We arithmetize the set theory of ]E , and apply Godel numbers r a ~ 1 cf)\"1 & IE f= 4> (a) . From (a) we haves Dwp (a) ++ IE \\= \u00a5x(. i|\u00bbtx) x = a ) S a t ( a 0 , I E ,a) , where a 0 = rVx ( i|\u00bb (x) -> x = a i s dependent on a, and IE i s f i x e d . Next, we look a t the c l a s s o f s e t s which are the v a l u e s o f a b s t r a c t i o n terms whose o n l y parameters are o r d i n a l s . T h i s c l a s s i s known i n the l i t e r a t u r e as the o r d i n a l - d e f i n -a b l e s e t s , denoted OD. Because o f P r o p o s i t i o n 6.1, the d e f i n a b i l i t y of t h i s c l a s s w i t h i n ZF must be shown w i t h an extended form of the r e f l e c t i o n p r i n c i p l e ( [ 1 5 ] f p. 273 ). Sin c e we are working w i t h i n 3ML [G] which has a s e t as u n i -v e r s e , we can use the s i m p l e r d e v i c e o f P r o p o s i t i o n 6.2(b). We f i r s t a r i t h m e t i z e the formulas of ZF, a s s i g n i n g them g unique Godel numbers as i n [24] , pp. 175 - 95. We l e t %| denoce the s e t of Godel numbers of formulas of ZF. ^ may be thought of as a countable s e t of o r d i n a l s i n M [ G ] . Our analogue to OD must i n c l u d e an a d d i t i o n a l parameter, namely a countable sequence of o r d i n a l s , so we g i v e i t a d i f f e r e n t symbol. D e f i n i t i o n 6.3 x e OD' M[G] h (3r' 107 (aa 9 \u2022 \u2022 '\u00b0n E Q M ) ( V Y ) C Y C * t, a. OD' i s the family of sets ordinal-definable i n M[G] from a sequence of ordinals. For each set of reals E e OD', Theorem 5.10 asserts that E i s Lebesgue measurable. Neither OD nor OD' are necessarily t r a n s i t i v e , as ele-ments of OD ( resp. OD' ) sets may not be OD ( resp. OD' ). OD does contain, however, a t r a n s i t i v e submodel known i n the l i t e r a t u r e as the family of he r e d i t a r i l y ordinal-def i n -able sets ( HOD ). The sets of HOD and a l l t h e i r ancestors v i a the membership r e l a t i o n are HOD. The t r a n s i t i v e closure TC(x) of a set x i s the least set containing x that i s t r a n s i t i v e . Existence of TC(x) presents no problem: Proposition 6.4 Let IE be a standard t r a n s i t i v e model of ZF. Vx e IE , TC (x) e IE Proof: Let the sequence (x } be defined: X 0 X, X n+1 = U x By standardness n and t r a n s i t i v i t y , along with the axioms of I n f i n i t y and ReplacementJ {x } e IE By the axiom of Regularity, u x = TC (x). . n n By the axiom of Unions, TC (x) e IE . These conditions are es s e n t i a l ; see [4], p. I l l 108 Just as, QD< js pur analogue to OD, our d e f i n i t i o n of UN i s analogous to that of HOD = { x e OD : TC (x) e OD } . D e f i n i t i o n 6.5 3N = f x e OD' ; TC (x) e OD' } . Lemma 6.6 1 = { x e OD' : x c I } Proof: By 6.4, TC(x) = {x} U { TC(.y) : y e x } , so x e EI i f f x e OD' & (Vy e x) (y e EJ ) . Corollary 6. 7 EI i s t r a n s i t i v e . Despite the close d e f i n i t i o n a l analogy between the class HOD and the set EI , there i s one important d i f f e r -ence. HOD s a t i s f i e s AC ( see [15], p. 276 ), but we s h a l l see that EI does not. The Myhill-Scott proof for HOD |= ZF adapts e a s i l y to EI . Theorem 6.8 EI |= ZF . Proof: (a) Since EI i s t r a n s i t i v e , the follow-ing hold ( see [9], pp. 21, 23 ): (i) EI i s extensional. ( i i ) (Xfy)1* = (x,y) . , \u2022 \u2022 \u2022 , EI ( i n ) u x = u x (iv) HP M (x) = 3P (x) H EI . These v e r i f y the axioms of Extensional-i t y , Pairs, Unions, and Powerset i n EI . Cb) EI i s standard and e n.EJ2 i s well-109 founded, so the axiom of Regularity holds i n IN , Cc) 0 e IN and to e IN , so the axioms of Null set and I n f i n i t y hold in IN . Cd) The axiom schema of Separation holds i n IN : (i) If x i s definable from parameters b^, ... ,b n which are i n OD', then x e OD'. This i s obvious from 6.3; the ordinal parameters for each b^ parametrize x, and the ordinal sequence parameters t ^ for each b^ can be amal-gamated into a single ordinal sequence parameter t for x, i n which the t ^ ap-pear as subsequences: = o 1 ^ t(k) = .t \u00b1 C j ) ; k = 2^3-0 ; otherwise ( i i ) Let f be a formula and l e t a,b, , ... ,b e IN cr OD' . Then: 1 n c = { x e a : IN \\= cf>(x,b^, . . . r^>n) } e OD' from (i) . c e IN by Lemma 6.6 . (e) The axiom schema of Replacement holds i n IN : Suppose: (i) f = { (x,y) e IN2 : IN [= i>(x,y,b1 , . . . ,b ) V where b, , ... ,b j_ n J 1 n 110. E I c OD' . ( i i ) JN |= f i s a f u n c t i o n . Then f o r a e JN , c = {y : JN f= (Vx e a )c|)(x,y,b 1 ). . ...,bn)'} belongs t o OD' , as i n the argument f o r (d). T h e r e f o r e c E I by Lemma 6.6 . From P r o p o s i t i o n 6 .2(b) and D e f i n i t i o n s 6.3 and 6.5,. we see t h a t Dwp(JN). The f o l l o w i n g u n i f o r m i t y r e s u l t p r e -sents t h i s f a c t i n a more s i g n i f i c a n t and u s e f u l form. Lemma 6.9 There i s a formula cj>0 f r e e i n s, y o n l y , such t h a t : 1 I\u2014 -u- r- TNT I I c c ^n, JM [G] h x e U (3s e W 0 W ) (Vy) ( 4> 0(s fy) y \u2022 = x ) . Proof : L e t : 6 0 ( s , y ) e\u00ab})(aa 1, ... , a n e 0 M ) (Vz) ([z e y +-* 0(y)) \u00bb \u00bb where 0 (y) -*->- TC(y) cr OD', which i s f r e e i n y o n l y ( when f u l l y w r i t t e n out i n ZF u s i n g 6.3 and 6.4 ). For any x e IN , Lemma 6.9 t e l l s us t h a t x i s u n i q u e l y determined by a sequence s e ^Q-^- \u2022 The next lemma ex-p l o i t s t h i s to show t h a t a l a r g e f a m i l y of mathematical o b j e c t s o f JM [G] e x i s t i n JN . I l l Lemma 6.10, Let h:u \u2022> JN and h e M [G] , then h e JN . Proof: Working i n M [ G ] , we def i n e an o r d i n a l y (x) as the l e a s t o r d i n a l a such that there i s an s:co -> a such t h a t x i s the unique y s a t i s f y i n g <)>o(s,y) (. Lemma 6.9 ). Let y = sup Y(.h(n)). Using AC, we define a n we 11-ordering L o n { s : S:OJ->Y } \u2022 Let s :&) Y be the l e a s t of these s such t h a t n ' h(n) i s the unique y s a t i s f y i n g 4> 0(s,y). We amalgamate the sequences s n i n t o a s i n g l e se-quence g:w -> 0 M : g(k) = \/ \\ i ~m_n s (n) ; k = 2 3 m ; otherwise h i s d e f i n a b l e from {s } v i a the w e l l - o r d e r i n g . n y = h(n) (Vs < s n) ( H K (s,y) ) & * Q ( s R , y ) , and {s n} i s c l e a r l y d e f i n a b l e from g. Thus, h e OD1. Since co e 3N and the axiom of P a i r s holds i n I , h c B by d e f i n i t i o n . h e 3N by Lemma 6.6 C o r o l l a r y 6.11 (a) IR M [ G ] e M . The above techniques are a l l we need to prove the l a s t two main theorems. Theorem 6.12 112 IN ^ DC Proof; Let A, R e I s a t i s f y the hypothesis of DC. Suppose {x^}^ \u00a3 n i s a f i n i t e se-quence of elements of A such that x.Rx.,, , Vi < n . Since M [G] |= AC, l l + l ' we may pick x , , e A such that x Rx ,. J ^ n+1 n n+1 for any value of n. By induction, there i s a map h:co -> A such that h(n) = x n . From Lemma 6.10, h e I . Theorem 6.13 I h LM . Proof: By Corollary 6.11, each r e a l i n M[G] belongs to IN , each closed i n t e r v a l with r a t i o n a l end-points i n IMC[G] belongs to IN , and each Borel code i n IMC [G] belongs to IN . Thus IMC [G] and IN have the same IN Borel sets. Let E cr IR , E e IN , then by d e f i n i t i o n of IN and Theorem 5.10 and Lemma 2.22, IMC [G] |= (E) +-> (3G (3N) ( y (N) = 0 & E = G^ -N ) . From Theorem 2.24 and the above: IN |= cj)(E) , i . e . , E i s Lebesgue measurable i n IN. From the assumption that there exists a model M e l 113 such, that M j= ZFC + I r we have arrived through these l a s t two sections at a model I e I s a t i s f y i n g 3N (= ZF + DC + LM . 114 Conclusion To summarize the development of the past seven sections, we w i l l present the main results i n the form of a theorem. Theorem Let IK be a non-minimal 1 standard t r a n s i t i v e model of ZFC + I . Then: (a) IK (= there i s a model of ZFC + \" Every set of reals definable from a countable seq-uence of ordinals i s Lebesgue measurable \" (b) IK f= there i s a model of ZF + DC + LM . We note that for IE = ( E ^ f t e ) , where E i s a set, and (j) a sentence, the statements IE |= ', and IE |= ZF, are ex-ZF pressible as A x -formulas ( [4], pp. 94, 96, 97 ). Hence the statements \" there exists a model of ... \" are abbrev-iation s of formulas i n the language of set theory. Our point of view i s c l e a r l y d i f f e r e n t from that of Solovay [23]. His model construction takes place within the i n t u i t i v e but ambiguous \"real world\" of set theory, while our constructions are r e l a t i v i z e d to a fixed model IK of set theory. For t h i s reason, Solovay's construction appears i n the form of a model extension. Our actual con-stuction takes the same form of model extension, but our models M, M [G] , and IN are inte r n a l models with respect to IK . This explains the format of our main theorem above. Theorems 0.36, 5.10, and 6.8, 6.12, 6.13 provide the i . e . IK i s not the least such model. See pp. 9 - 1 0 . 115 main v e r i f i c a t i o n of t h i s theorem, except for the found-ation a l aspects to which we now return. Axiom A, which worked so well as a foundation for Sec-tions 0 - 3 , proved not to be powerful enough ( p.91 ) to ensure the existence of an inaccessible i n the ground model. Even the adoption of the much stronger Axiom I as a part of our metatheory would f a i l to do t h i s . Perhaps the best com-promise would be to introduce an axiom that guarantees the existence of at least two inaccessibles. Standard techniques ( e.g. [4], p.110 ) then produce a set K such that IK \\= I . Now, supposing that K does e x i s t such that IK \\= I, we must s t i l l produce a countable ground model M s a t i s f y i n g I and belonging to IK . A standard modification of the Lowenheim-Skolem-Tarski technique exists by which we can constuct a countable elem-entary submodel belonging to IK , ( IK i s non-minimal ) , which we outline ( see [12], c.f. [4] ). F i r s t we take the c l o -sure of {0} under a set of Skolem functions for IK and Mostowski-collapse t h i s to a t r a n s i t i v e set M. M w i l l be a countable elementary submodel of IK , thus M (= I. By a theorem of Levy ( [4], p.104 ) M i s h e r e d i t a r i l y countable ( |TC(M)| < O)I ) which implies that M has countable rank-( [4] , p. 103 ) . Hence M c { x e l : rank(x) < to i} e IK . The power set axiom then t e l l s us that M e l . This completes the proof of our theorem. 116 Bibliography C. C. Chang and J. H. Kei s l e r , [1] Model Theory , ( North-Holland, Amsterdam, 1973 ). P. J. Cohen, [2] Independence res u l t s i n set theory, i n : J . W. Addison, L. Henkin, and A. Tarski, eds., The Theory of Models, ( North-Holland, Amsterdam, 1965 ) 39 - 54. 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