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Notes on Forcing Notes on forcing M. Viale, with the collaboration of F. Calderoni, and R.Carroy 2 Contents 1 Introduction1 2 Boolean algebras9 2.1 Orders and topology...........................9 2.1.1 Orders...............................9 2.1.2 Topological spaces........................ 11 2.2 Boolean algebras............................. 14 2.2.1 The order on boolean algebras.................. 15 2.2.2 Boolean identities......................... 16 2.2.3 Ideals and morphisms of boolean algebras........... 17 2.2.4 Atomic and finite boolean algebras............... 18 2.2.5 Examples of boolean algebras.................. 20 2.2.6 The Prime Ideal Theorem.................... 23 2.2.7 Stone spaces of boolean algebras................. 25 2.2.8 Boolean rings........................... 28 2.2.9 Boolean algebras as complemented distributive lattices.... 31 2.2.10 Suprema and infima of subsets of a boolen algebra....... 32 2.3 Complete boolean algebras........................ 34 2.3.1 Complete boolean algebras of regular open sets........ 35 2.3.2 Boolean completions....................... 40 2.4 Some remarks on partial orders and their boolean completions.... 45 2.5 Stone-Cech compactifications...................... 45 3 Combinatorial properties of partial orders 53 3.1 Filters, antichains, and predense sets on partial orders......... 53 3.2 The posets F n(X; Y )........................... 57 3.2.1 The poset F n(!2 × !; 2)..................... 59 3.3 Pre-orders with the countable chain condition............. 59 3.3.1 The algebra of Lebesgue measurable sets modulo null sets... 62 4 Boolean Valued Models 65 4.1 Boolean valued models and boolean valued semantics......... 66 4.2 Examples of boolean valued models................... 71 4.3 Full boolean valued models andLo´stheorem .............. 75 4.3.1Lo´stheorem for full boolean valued models........... 75 4.3.2 Examples of full boolean valued models I: spaces of measurable functions.............................. 78 i ii CONTENTS 4.3.3 Examples of full boolean valued models II: standard ultra- products.............................. 79 4.3.4 Examples of full boolean valued models III: C(St(B); 2!)... 80 4.4 Full B-models for a non complete boolean algebra B .......... 81 5 Forcing 83 5.1 Boolean valued models for set theory.................. 85 5.1.1 External definition of forcing................... 87 5.1.2 Internal definition of forcing: the boolean valued semantic of V B ................................. 98 5.2 Cohen's forcing theorem......................... 109 5.2.1 How to use Cohen's forcing theorem.............. 114 5.3 Independence of CH ............................ 116 5.3.1 A model of :CH ......................... 116 5.3.2 A model of CH .......................... 121 5.4 M B models ZFC .............................. 126 6 Appendix 131 6.1 Absoluteness................................ 131 6.2 Syntax and semantics inside V ...................... 137 6.2.1 Syntax............................... 137 6.2.2 Semantics............................. 138 6.2.3 Getting countable transitive models of ZFC ........... 139 Chapter 1 Introduction These notes are meant for students in mathematics who wish to follow the set theory course in the master program of Torino university. The course program covers large portions of Kunen's book [5] (or its new edition [6]). We focus in particular on: • Axiomatic set theory, including basics of cardinal and ordinal arithmetic, transfinite recursion and the Mostowski collapsing theorem, the reflection the- orems, the absoluteness properties of transitive structures, the development of basic model theory (including the L¨owenheim-Skolem theorem for set struc- tures) inside the standard model V for ZFC. This material is extensively covered in [5, Chapters I,III,IV,V] or [6, Chapter I, and Sections II.1{II.5]. • Forcing and combinatorics of partial orders ( [5, Chapters II,VII] or [6, Chap- ters III, IV]). Since the course presents this material in a way which differs substantially from the approaches taken in [5] or [6], we decided to write up these notes to cover this part of the program. These notes are divided in five chapters and an appendix. The second and third recall standard material on partial orders, boolean algebras, and topological spaces which can be found in several textbooks. The fourth introduces boolean valued semantics as a natural generalization of Tarski semantics for first order logic. The fifth chapter is the heart of these notes. It gives a detailed presentation of the forcing method via boolean valued models. It features the most celebrated application of this method, namely the undecidability of the continuum hypothesis CH. The appendix gives some more details on the parts of our course for which the reference texts are [5, Chapters III,IV,V] or [6, Chapter I, and Sections II.1{II.5]. The reader already familiar with the material covered in the first two chapters of these notes may just skim through this part or refer back to it if needed while reading the third and fourth chapter. • Chapter2 introduces basic notions regarding partial orders, topological spaces, and boolean algebras. Some basic properties of the dualities which link these apparently distinct categories of mathematical objects are recalled. We prove in particular that every partial order admits, up to isomorphism, a unique complete boolean algebra in which it embeds as a dense subset, and we link the basic theory of boolean algebras to that of compact Haussdorff spaces 1 2 CHAPTER 1. INTRODUCTION by means of the Stone representation theorem. We bring to the attention of the reader that this first chapter presents material on boolean algebras and compact spaces rather standard which she/he may already have encountered in other courses. We decided to include it here in order to present these results with a focus aimed towards their use within set theory and their role in the development of the forcing method. • Chapter3 introduces some basic combinatorial properties of partial orders which will be needed to prove the independence of the continuum hypothesis by means of the forcing method. In particular we focus on CCC partial orders, we prove the ∆-system Lemma, and we use it to prove that the notion of forcing which can be used to produce a model of ZFC where CH fails is CCC. Most (if not all) of these results can be found in [6, III.1-III.2-III.3] or [5, II] , however these sections of both books contain a large amount of material which is not strictly necessary to present the two basic applications of the forcing method we aim for in these notes. The role of this chapter is to collect the minimal amount of information needed to run our applications of the forcing method. We also aim to outline the connections between the combinatorial properties of certain partial orders and well known topological properties of the Stone spaces associated to them (among other things we prove the Baire category theorem for compact Hausdorff spaces). • Chapter4 introduces the boolean valued semantics for first order theories and shows that certain function spaces which naturally occur in functional analysis produce natural examples of boolean valued models. The boolean valued semantic selects a given complete boolean algebra B and assigns to every statement φ a boolean value in B. The boolean operations reflect the behavior of the propositional connectives; it requires more attention to give a meaning to atomic formulae and to quantifiers, and we need a certain amount of completeness for B in order to be able to interpret quantifiers in the boolean semantic. • Chapter5 develops the theory of forcing by means of boolean valued models giving detailed proofs of the basic properties of boolean valued models for set theory, of Cohen's forcing theorem, and the two basic applications of the method which suffice to prove the independence of CH from the standard axioms of set theory. Our presentation of forcing departs completely from the approach taken by Kunen and is more keen to that taken by Bell [2],Jech [4]. We decide to follow this different approach for two reasons: 1. In our eyes, the boolean valued models approach makes the metamath- ematical arguments needed to understand the forcing method easier to grasp and greatly simplifies some proofs. 2. The boolean valued model approach makes more transparent what is the role played by generic filters in the development of the forcing method and where the hypothesis that a filter is suitably generic is essential. Moreover it enlightens the link existing between the notion of generic filter arising 3 in forcing with the corresponding topological notion of generic point of a topological space which is at the heart of the Baire category arguments. Typographical conventions All over these notes in some occasions we introduce some arguments and material which are not central for the development of the core results. We adopt the typographical convention to put these parts of our notes in a smaller font. Prerequisites. We assume that the reader of these notes has familiarity with the content of [5, Chapter I, Sections II.1{II.5] or [6, Chapter III,IV,V] (see the appendix for more details). This familiarity is not of vital importance for the comprehension of the first four chapters of these notes, where just a basic knowledge of the axioms of set theory is required and no familiarity with the formal first order development of ZFC is needed. On the other hand to understand the fifth chapter on forcing the reader must know: • the basic facts about cardinal arithmetic and well orders (what
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