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Generic Large Cardinals and Absoluteness

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Generic Large Cardinals and Absoluteness View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Institutional Research Information System University of Turin Universita` degli Studi di Torino Facolta` di Scienze M.F.N. Dipartimento di Matematica \G. Peano" Generic Large Cardinals and Absoluteness Giorgio Audrito Dottorato in Matematica { XXVII ciclo Tutor: Prof. Matteo Viale Coordinatore: Prof. Ezio Venturino Anno Accademico 2015/2016 Settore Scientifico Disciplinare MAT/01 CONTENTS Introduction iii Notation . viii 1 Backgrounds 1 1.1 Second-order elementarity and class games . .1 1.2 Generalized stationarity . .3 1.3 Forcing and boolean valued models . .9 2 Iteration systems 13 2.1 Embeddings and retractions . 13 2.1.1 Embeddings and boolean valued models . 17 2.2 Iteration systems . 18 2.2.1 Boolean algebra operations on iteration system limits . 19 2.2.2 Relation between inverse and direct limits . 21 2.3 Two-step iterations and generic quotients . 22 2.3.1 Two-step iterations . 23 2.3.2 Generic quotients . 25 2.3.3 Equivalence of two-step iterations and regular embeddings . 27 2.3.4 Generic quotients of iteration systems . 28 3 Forcing axioms 31 3.1 Forcing classes . 32 3.1.1 Algebraic formulation of properness and semiproperness . 33 3.2 Forcing axioms . 35 3.2.1 Forcing axioms as density properties . 38 3.2.2 Resurrection axioms . 39 3.3 Weak iterability . 39 3.4 Semiproper and SSP iterations . 42 3.4.1 Semiproper two-step iterations . 42 3.4.2 Semiproper iteration systems . 45 3.4.3 Stationary set preserving iterations . 49 i 4 Generic absoluteness 51 4.1 Iterated resurrection and absoluteness . 51 4.1.1 Resurrection game . 53 4.1.2 Resurrection axioms and generic absoluteness . 57 4.2 Uplifting cardinals and definable Menas functions . 58 4.2.1 Consistency strength of (α)-uplifting cardinals . 59 4.2.2 Reflection properties of (α)-uplifting cardinals . 60 4.2.3 Variations and strengthenings of (α)-upliftingness . 60 4.2.4 Menas functions for uplifting cardinals . 61 4.3 Consistency strength . 62 4.3.1 Upper bounds . 63 4.3.2 Lower bounds . 68 4.4 Conclusions and open problems . 70 5 Systems of filters and generic large cardinals 73 5.1 Systems of filters . 73 5.1.1 Main definitions . 74 5.1.2 Standard extenders and towers as C-systems of filters . 77 5.1.3 Systems of filters in V and generic systems of ultrafilters . 80 5.1.4 Embedding derived from a system of ultrafilters . 82 5.1.5 System of ultrafilters derived from an embedding . 85 5.2 Generic large cardinals . 89 5.2.1 Deriving large cardinal properties from generic systems of filters 91 5.2.2 Consistency of small generic large cardinals . 93 5.2.3 Combinatorial equivalents of ideally large cardinal properties 97 5.2.4 Distinction between generic large cardinal properties . 103 5.3 Derived ideal towers in V failing preservation of large cardinal properties104 5.4 Conclusions and open problems . 108 ii INTRODUCTION In the wake of early G¨odel'sIncompleteness Theorems, a large part of set theory has revolved around the study of the independence phenomena. The most significant boost in this field was due to the introduction of the forcing technique. Many inde- pendence results were found in different areas of mathematics since then, including among many the continuum hypothesis [32], Borel's conjecture [34], Kaplansky's conjecture [14], Whitehead problem [38], strong Fubini theorem [18], and the exis- tence of outer automorphisms of the Calkin algebra [15]. A dual approach in set theory is to search for new axioms of mathematics in order to control the independence phenomena. These new axioms need to be justified, so that their mathematical properties interact with philosophical guidelines. An axiom candidate can be judged both from its premises, that is, whether the statement can be argued to be a \natural" extension of the accepted axioms of ZFC, and from its consequences, that is, whether the statement is able to entail a large number of desired properties in mathematics. In this thesis we follow this approach, considering axioms in two of the most important families: forcing axioms (Chapters 3, 4) and large cardinals (Chapter 5), considering their effectiveness both on the premises and consequences side. While these chapters follow mostly independent paths, all of them fit in the same quest for natural and powerful axioms for mathematics, and are strongly dependent on the iterated forcing technique covered in Chapter 2. Basic notation and background topics are dealt with in Chapter 1. In all the chapters of this thesis we shall focus on the boolean valued models approach to forcing. A partial order and its boolean completion can produce exactly the same consistency results, however: • in a specific consistency proof the forcing notion we have in mind in order to obtain the desired result is given by a partial order and passing to its boolean completion may obscure our intuition on the nature of the problem and the combinatorial properties we wish our partial order to have; • when the problem aims to find general properties of forcings which are shared by a wide class of partial orders, we believe that focusing on complete boolean algebras gives a more efficient way to handle the problem. In fact, the rich algebraic theory of complete boolean algebras can greatly simplify our calcu- lations. iii 0 Introduction Since the focus of the thesis is on the general correlations between forcing, large cardinals, and forcing axioms, rather than on specific applications of the forcing method to prove the consistency of a given mathematical statement, we are naturally led to analyze forcing through an approach based on boolean valued models. Iterated forcing is a procedure that describes transfinite applications of the forc- ing method, with a special attention to limit stages, and is one of the main tools for proving the consistency of forcing axioms. In Chapter 2 we give a detailed account of iterated forcing through boolean algebras, inspired by the algebraic approach of Donder and Fuchs [19], thus providing a solid background on which the rest of the thesis is built. All the material in this chapter is joint work with Matteo Viale and Silvia Steila. Even though all the results in this chapter come from a well established part of the current development of set theory, the proofs are novel and (according to us) simpler and more elegant, due to a systematic presentation of the whole theory of iterated forcing in terms of algebraic constructions. In Chapter 3 we introduce the notion of weakly iterable forcing class and prove the preservation theorems for semiproper and stationary set preserving iterations (for the latter result, we assume the existence of class many supercompact cardinals). We believe there will be no problem in rearranging these techniques in order to cover also the cases of proper or ccc iterations. Forcing axioms are set-theoretic principles that arise directly from the technique of forcing. It is a matter of fact that forcing is one of the most powerful tools to produce consistency results in set theory: forcing axioms turn it into a powerful instrument to prove theorems. This is done by showing that a statement φ follows from an extension T of ZFC if and only if T proves that φ is consistent by means of a certain type of forcing. These types of results are known in the literature as generic absoluteness results and have the general form of a completeness theorem for some T ⊇ ZFC with respect to the semantic given by boolean valued models and first order calculus. More precisely generic absoluteness theorems fit within the following general framework: Assume T is an extension of ZFC, Θ is a family of first order formulas in the language of set theory and Γ is a certain class of forcing notions definable in T . Then the following are equivalent for a φ 2 Θ and S ⊇ T : 1. S proves φ. 2. S proves that there exists a forcing B 2 Γ such that B forces φ and T jointly. 3. S proves that B forces φ for all forcings B 2 Γ such that B forces T . We say that a structure M definable in a theory T is generically invariant with respect to forcings in Γ and parameters in X ⊂ M when the above situation occurs with Θ being the first order theory of M with parameters in X. A brief overview of the main known generic absoluteness results is the following: iv • Shoenfield’s absoluteness theorem is a generic absoluteness result for Θ the 1 family of Σ2-properties with real parameters, Γ the class of all forcings, T = ZFC. • The pioneering \modern" generic absoluteness results are Woodin's proofs of the invariance under set forcings of the first order theory of L(ON!) with real parameters in ZFC +class many Woodin cardinals which are limit of Woodin cardinals [33, Thm. 3.1.2] and of the invariance under set forcings of the 2 family of Σ1-properties with real parameters in the theory ZFC + CH +class many measurable Woodin cardinals [33, Thm. 3.2.1]. Further results pin down the exact large cardinal strength of the assertion that L(R) is generically invariant with respect to certain classes of forcings (among others see [37]). • The bounded forcing axiom BFA(Γ) is equivalent to the statement that generic absoluteness holds for T = ZFC and Θ the class of Σ1-formulas with parame- ters in P(!1), as shown in [6]. • Recently, Hamkins and Johnstone [23] introduced the resurrection axioms RA(Γ) and Viale [42] showed that these axioms produce generic absoluteness for Θ the Σ2-theory with parameters of Hc, T = ZFC + RA(Γ), Γ any of the standard classes of forcings closed under two step iterations.
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