DOCSLIB.ORG

The Envelope of a Pointclass Under a Local Determinacy Hypothesis

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Envelope of a Pointclass Under a Local Determinacy Hypothesis THE ENVELOPE OF A POINTCLASS UNDER A LOCAL DETERMINACY HYPOTHESIS TREVOR M. WILSON Abstract. Given an inductive-like pointclass Γ and assuming the Ax- iom of Determinacy, Martin identified and analyzede the pointclass con- taining the norm relations of the next semiscale beyond Γ, if one exists. We show that much of Martin's analysis can be carriede out assuming only ZF + DCR + Det(∆Γ). This generalization requires arguments from Kechris{Woodin [10] ande e Martin [13]. The results of [10] and [13] can then be recovered as immediate corollaries of the general analysis. We also obtain a new proof of a theorem of Woodin on divergent models of AD+, as well as a new result regarding the derived model at an inde- structibly weakly compact limit of Woodin cardinals. Introduction Given an inductive-like pointclass Γ, Martin introduced a pointclass (which we call the envelope of Γ, following [22])e whose main feature is that it con- tains the prewellorderingse of the next scale (or semiscale) beyond Γ, if such a (semi)scale exists. This work was distributed as \Notes on the nexte Suslin cardinal," as cited in Jackson [3]. It is unpublished, so we use [3] as our reference instead for several of Martin's arguments. Martin's analysis used the assumption of the Axiom of Determinacy. We reformulate the notion of the envelope in such a way that many of its essen- tial properties can by derived without assuming AD. Instead we will work in the base theory ZF+DCR for the duration of the paper, stating determinacy hypotheses only when necessary. With the exception of Section 9 this paper is mainly expository. It places results of Martin, Kechris, Woodin, and others in a unified framework and in some cases substantially simplifies the original proofs. Section 1 contains background information on descriptive set theory in- cluding the definition of \inductive-like pointclass." In Section 2 we define the envelope of an inductive-like pointclass in a special case that will suffice for results in L(R) and we prove some results about it, most notably its determinacy and its closure under real quantifiers. In Sections 3 and 4 we apply these results to the envelope of the pointclass of inductive sets and Date: May 4, 2014. 2010 Mathematics Subject Classification. Primary 03E15; Secondary 03E60. Key words and phrases. determinacy, pointclass, scale, derived model. The author gratefully acknowledges support from NSF grant DMS-1044150. 1 2 TREVOR M. WILSON then more generally to the envelopes of inductive-like pointclasses in L(R). In particular we show that the Kechris{Woodin determinacy transfer theo- rem and a theorem of Martin on reflection of ordinal-definability for reals can be obtained as immediate consequences. In Section 6 we give a more general definition of the envelope for boldface inductive-like pointclasses, which is phrased in terms of Moschovakis's notion of \companion" for such pointclasses (see Section 5.) We then show that the results of Section 2 can readily be adapted to this more general setting. In Section 7 we give an equivalent condition for the existence of semiscales that is phrased in terms of games. In the later sections these games are used to construct (under additional assumptions) semiscales on a universal Γˇ set with prewellorderings in the envelope of Γ. In Section 8 we give a new proofe of Woodin's theorem on divergent modelse of AD+. Finally, in Section 9 we show that the derived model at an indestructibly weakly compact limit of Woodin cardinals satisfies \every set of reals is Suslin." The results in this paper are mostly taken from the author's PhD thesis [25, Ch. 3]. The author wishes to thank John Steel, who supervised this thesis work, for his guidance; Hugh Woodin, for explaining his argument for the result stated as Theorem 8.2 below; and Martin Zeman, for suggesting several corrections. 1. Pointclasses We begin by recalling some standard notions from descriptive set theory. ! By convention we denote the Baire space ! by R and call its elements \reals." A product space is a space of the form X = X1 × · · · × Xn;Xi = R or Xi = ! for all i ≤ n: A pointset is a subset of a product space. A pointclass is a collection of pointsets, typically an initial segment of some complexity hierarchy for pointsets. We say that a pointclass Γ is !-parameterized (respectively, R-parameter- ized) if for every product space X there is a Γ subset of !×X (respectively, of R × X ) that is universal for Γ subsets of X . In this paper we consider the following types of pointclasses, which are named for their resemblance to the pointclasses IND and IND of induc- tive sets and boldface inductive sets respectively. The inductive] pointsets are those that are definable without parameters by positive elementary in- duction on R (sometimes called \absolutely" or \lightface" inductive.) By \boldface" we mean that real parameters are allowed. Definition 1.1. A pointclass Γ is (lightface) inductive-like if it is !-parameterized, closed under 9R, 8R, and recursive substitution, and has the prewellordering property. THE ENVELOPE UNDER LOCAL DETERMINACY 3 Definition 1.2. A pointclass Γ is boldface inductive-like if it is R-parameterized, closed under 9R, 8R, and continuouse reducibility, and has the prewellorder- ing property. Note that some authors strengthen the prewellordering property to the scale property in these definitions. In this paper we will typically use the following notational conventions. By Γ and Γ we will denote an inductive-like or boldface-inductive-like pointclass respectively.e As usual we denote the dual pointclass of Γ by Γˇ = f:A : A 2 Γg and the ambiguous part of Γ by ∆Γ = Γ \ Γ.ˇ (Note that the existence of universal sets implies non-self-duality by a diagonal argument.) Similarly for a boldface pointclass we define the dual and ambiguous pointclasses Γˇ and ∆Γ. e Givene e a lightface pointclass Γ we can define the corresponding relativiza- tions Γ(x) for x 2 and also the boldface pointclasses Γ = S Γ(x), R x2R Γˇ = S Γ(ˇ x), and ∆ = S ∆ (x). Naturally, if Γe is inductive-like x2R Γ x2R Γ thene the correspondinge boldface pointclass Γ is boldface inductive-like. Typically given a pointclass Γ we will assumee the local determinacy hypo- thesis Det(∆Γ), meaning that every two-player game of perfect information on ! with ae∆Γ payoff set is determined, although some of our results can proved undere weaker hypotheses. The prewellordering ordinal of an inductive-like pointclass Γ (or more generally of a pointclass Γ with the prewellordering property) is the supre- mum of the ordinal lengths of all ∆Γ prewellorderings of R, or equivalently the range of any regular Γ-norm one a complete Γ pointset. We will often denote the prewellordering ordinal of a pointclass by κ. The prototypical example of an inductive-like pointclass is, as noted above, the pointclass IND of inductive sets. Other examples of inductive-like 2 L(R) pointclasses include the pointclass (Σ1) and more generally the point- 2 + class Σ1 under AD . Examples of boldface inductive-like pointclasses include, in addition to the boldface versions of the above lightface pointclasses, the pointclass S(κ) of κ-Suslin sets when AD holds, κ is a Suslin cardinal, and S(κ) is closed under 8R. The following notion of countable approximation for pointsets is central to the paper. Definition 1.3 (Martin, see [3, Defn. 3.9]). Let X be a product space, let κ be an ordinal, and let (Aα : α < κ) be a sequence of subsets of X . For a pointset A ⊂ X , we say A 2 (Aα : α < κ) if for every countable set σ ⊂ X there is an ordinal α < κ such that A \ σ = Aα \ σ. Martin then made the following definition. It is similar to the definition of \envelope" that we will use in this paper, but we will need to make a significant modification. We include the original definition below in order to provide historical context. 4 TREVOR M. WILSON Definition 1.4 (Martin, see [3, Defn. 3.9]). Let ∆ be a pointclass and let κ be an ordinal. For a product space X and a pointsete A ⊂ X , we say A 2 Λ(∆; κ) if A 2 (Aα : α < κ) for some sequence (Aα : α < κ) of ∆ subsets ofe X . e If AC holds then the pointclass Λ(∆; κ) is not likely to be very useful because there will be too many well-orderede sequences of pointsets. For example, if ∆ is a nontrivial boldface pointclass (so it contains every count- able pointset)e and κ ≥ c then any pointset whatsoever is in Λ(∆; κ). To get a useful definition without assuming AD we need to require somee kind of uniformity for the sequence of pointsets (Aα : α < κ). First we deal with a special case. Jκ(R) 2. The envelope of Σ1 e We will need some background on the fine structure of L(R) that can be found, for example, in [23, x1]. We use the Jensen hierarchy (Jα : α 2 Ord) for L(R) but the reader would not miss much by reading \Lκ(R)" in place of \Jκ(R)." However we should note that the ordinal height of Jα(R) is !α rather than α. By convention we allow R as an unstated parameter in our formulas. For example, by Σ1 we mean Σ1(fRg). Therefore quantification over the reals always counts as bounded quantification. R Given ordinals κ and β with κ ≤ β, we say Jκ(R) ≺1 Jβ(R) if Jκ(R) is a Σ1(R [ fRg)-elementary substructure of Jβ(R).
Load more

Recommended publications
  • Calibrating Determinacy Strength in Levels of the Borel Hierarchy
    CALIBRATING DETERMINACY STRENGTH IN LEVELS OF THE BOREL HIERARCHY SHERWOOD J. HACHTMAN Abstract. We analyze the set-theoretic strength of determinacy for levels of the Borel 0 hierarchy of the form Σ1+α+3, for α < !1. Well-known results of H. Friedman and D.A. Martin have shown this determinacy to require α+1 iterations of the Power Set Axiom, but we ask what additional ambient set theory is strictly necessary. To this end, we isolate a family of Π1-reflection principles, Π1-RAPα, whose consistency strength corresponds 0 CK exactly to that of Σ1+α+3-Determinacy, for α < !1 . This yields a characterization of the levels of L by or at which winning strategies in these games must be constructed. When α = 0, we have the following concise result: the least θ so that all winning strategies 0 in Σ4 games belong to Lθ+1 is the least so that Lθ j= \P(!) exists + all wellfounded trees are ranked". x1. Introduction. Given a set A ⊆ !! of sequences of natural numbers, consider a game, G(A), where two players, I and II, take turns picking elements of a sequence hx0; x1; x2;::: i of naturals. Player I wins the game if the sequence obtained belongs to A; otherwise, II wins. For a collection Γ of subsets of !!, Γ determinacy, which we abbreviate Γ-DET, is the statement that for every A 2 Γ, one of the players has a winning strategy in G(A). It is a much-studied phenomenon that Γ -DET has mathematical strength: the bigger the pointclass Γ, the stronger the theory required to prove Γ -DET.
    [Show full text]
  • Effective Cardinals of Boldface Pointclasses
    EFFECTIVE CARDINALS OF BOLDFACE POINTCLASSES ALESSANDRO ANDRETTA, GREG HJORTH, AND ITAY NEEMAN Abstract. Assuming AD + DC(R), we characterize the self-dual boldface point- classes which are strictly larger (in terms of cardinality) than the pointclasses contained in them: these are exactly the clopen sets, the collections of all sets of ξ ξ Wadge rank ≤ ω1, and those of Wadge rank < ω1 when ξ is limit. 1. Introduction A boldface pointclass (for short: a pointclass) is a non-empty collection Γ of subsets of R such that Γ is closed under continuous pre-images and Γ 6= P(R). 0 0 0 Examples of pointclasses are the levels Σα, Πα, ∆α of the Borel hierarchy and the 1 1 1 levels Σn, Πn, ∆n of the projective hierarchy. In this paper we address the following Question 1. What is the cardinality of a pointclass? Assuming AC, the Axiom of Choice, Question 1 becomes trivial for all pointclasses Γ which admit a complete set. These pointclasses all have size 2ℵ0 under AC. On the other hand there is no obvious, natural way to associate, in a one-to-one way, 0 an open set (or for that matter: a closed set, or a real number) to any Σ2 set. This 0 0 suggests that in the realm of definable sets and functions already Σ1 and Σ2 may have different sizes. Indeed the second author in [Hjo98] and [Hjo02] showed that AD + V = L(R) actually implies 0 0 (a) 1 ≤ α < β < ω1 =⇒ |Σα| < |Σβ|, and 1 1 1 (b) |∆1| < |Σ1| < |Σ2| < .
    [Show full text]
  • The Realm of Ordinal Analysis
    The Realm of Ordinal Analysis Michael Rathjen Department of Pure Mathematics, University of Leeds Leeds LS2 9JT, United Kingdom Denn die Pioniere der Mathematik hatten sich von gewissen Grundlagen brauchbare Vorstellungen gemacht, aus denen sich Schl¨usse,Rechnungsarten, Resultate ergaben, deren bem¨achtigten sich die Physiker, um neue Ergebnisse zu erhalten, und endlich kamen die Techniker, nahmen oft bloß die Resultate, setzten neue Rechnungen darauf und es entstanden Maschinen. Und pl¨otzlich, nachdem alles in sch¨onste Existenz gebracht war, kamen die Mathematiker - jene, die ganz innen herumgr¨ubeln, - darauf, daß etwas in den Grundlagen der ganzen Sache absolut nicht in Ordnung zu bringen sei; tats¨achlich, sie sahen zuunterst nach und fanden, daß das ganze Geb¨audein der Luft stehe. Aber die Maschinen liefen! Man muß daraufhin annehmen, daß unser Dasein bleicher Spuk ist; wir leben es, aber eigentlich nur auf Grund eines Irrtums, ohne den es nicht entstanden w¨are. ROBERT MUSIL: Der mathematische Mensch (1913) 1 Introduction A central theme running through all the main areas of Mathematical Logic is the classification of sets, functions or theories, by means of transfinite hierarchies whose ordinal levels measure their ‘rank’ or ‘complexity’ in some sense appropriate to the underlying context. In Proof Theory this is manifest in the assignment of ‘proof the- oretic ordinals’ to theories, gauging their ‘consistency strength’ and ‘computational power’. Ordinal-theoretic proof theory came into existence in 1936, springing forth from Gentzen’s head in the course of his consistency proof of arithmetic. To put it roughly, ordinal analyses attach ordinals in a given representation system to formal theories.
    [Show full text]
  • Determinacy and Large Cardinals
    Determinacy and Large Cardinals Itay Neeman∗ Abstract. The principle of determinacy has been crucial to the study of definable sets of real numbers. This paper surveys some of the uses of determinacy, concentrating specifically on the connection between determinacy and large cardinals, and takes this connection further, to the level of games of length ω1. Mathematics Subject Classification (2000). 03E55; 03E60; 03E45; 03E15. Keywords. Determinacy, iteration trees, large cardinals, long games, Woodin cardinals. 1. Determinacy Let ωω denote the set of infinite sequences of natural numbers. For A ⊂ ωω let Gω(A) denote the length ω game with payoff A. The format of Gω(A) is displayed in Diagram 1. Two players, denoted I and II, alternate playing natural numbers forming together a sequence x = hx(n) | n < ωi in ωω called a run of the game. The run is won by player I if x ∈ A, and otherwise the run is won by player II. I x(0) x(2) ...... II x(1) x(3) ...... Diagram 1. The game Gω(A). A game is determined if one of the players has a winning strategy. The set A is ω determined if Gω(A) is determined. For Γ ⊂ P(ω ), det(Γ) denotes the statement that all sets in Γ are determined. Using the axiom of choice, or more specifically using a wellordering of the reals, it is easy to construct a non-determined set A. det(P(ωω)) is therefore false. On the other hand it has become clear through research over the years that det(Γ) is true if all the sets in Γ are definable by some concrete means.
    [Show full text]
  • Effective Descriptive Set Theory
    Effective Descriptive Set Theory Andrew Marks December 14, 2019 1 1 These notes introduce the effective (lightface) Borel, Σ1 and Π1 sets. This study uses ideas and tools from descriptive set theory and computability theory. Our central motivation is in applications of the effective theory to theorems of classical (boldface) descriptive set theory, especially techniques which have no classical analogues. These notes have many errors and are very incomplete. Some important topics not covered include: • The Harrington-Shore-Slaman theorem [HSS] which implies many of the theorems of Section 3. • Steel forcing (see [BD, N, Mo, St78]) • Nonstandard model arguments • Barwise compactness, Jensen's model existence theorem • α-recursion theory • Recent beautiful work of the \French School": Debs, Saint-Raymond, Lecompte, Louveau, etc. These notes are from a class I taught in spring 2019. Thanks to Adam Day, Thomas Gilton, Kirill Gura, Alexander Kastner, Alexander Kechris, Derek Levinson, Antonio Montalb´an,Dean Menezes and Riley Thornton, for helpful conversations and comments on earlier versions of these notes. 1 Contents 1 1 1 1 Characterizing Σ1, ∆1, and Π1 sets 4 1 1.1 Σn formulas, closure properties, and universal sets . .4 1.2 Boldface vs lightface sets and relativization . .5 1 1.3 Normal forms for Σ1 formulas . .5 1.4 Ranking trees and Spector boundedness . .7 1 1.5 ∆1 = effectively Borel . .9 1.6 Computable ordinals, hyperarithmetic sets . 11 1 1.7 ∆1 = hyperarithmetic . 14 x 1 1.8 The hyperjump, !1 , and the analogy between c.e. and Π1 .... 15 2 Basic tools 18 2.1 Existence proofs via completeness results .
    [Show full text]
  • A Classification of Projective-Like Hierarchies in L(R)
    A classification of projective-like hierarchies in L(R) Rachid Atmai Universit¨atWien Kurt G¨odelResearch Center W¨ahringerStraße 25 1090 Vienna [email protected] Steve Jackson Dept. of Mathematics General Academics Building 435 1155 Union Circle 311430 University of North Texas Denton, TX 76203-5017 [email protected] October 10, 2016 Abstract We provide a characterization of projective-like hierarchies in L(R) in the context of AD by combinatorial properties of their associated Wadge ordinal. In the second chapter we concentrate on the type III case of the analysis. In the last chapter, we provide a characterization of type IV projective-like hierarchies by their associated Wadge ordinal. 1 Introduction 1.1 Outline The paper is roughly organized as follows. We first review basic definitions of the abstract theory of pointclasses. In the second section we study the type III pointclasses and the clo- sure properties of the Steel pointclass generated by a projective algebra Λ in the case where cof(o(Λ)) > !. In particular we show that Steel's conjecture is false. This leads to isolat- ing a combinatorial property of the Wadge ordinal of the projective algebra generating the 1 Steel pointclass. This combinatorial property ensures closure of the Steel pointclass under dis- junctions. Therefore it is still possible to have κ regular, where κ is the Wadge ordinal for a projective algebra Λ generating the next Steel pointclass and yet the Steel pointclass is not closed under disjunction. In the third section, we look at the type IV projective-like hierarchies, that is the pointclass which starts the new hierarchy is closed under quantifiers.
    [Show full text]
  • Souslin Quasi-Orders and Bi-Embeddability of Uncountable
    Souslin quasi-orders and bi-embeddability of uncountable structures Alessandro Andretta Luca Motto Ros Author address: Universita` di Torino, Dipartimento di Matematica, Via Carlo Alberto, 10, 10123 Torino, Italy E-mail address: [email protected] Universita` di Torino, Dipartimento di Matematica, Via Carlo Alberto, 10, 10123 Torino, Italy E-mail address: [email protected] arXiv:1609.09292v2 [math.LO] 18 Mar 2019 Contents 1. Introduction 1 2. Preliminaries and notation 14 3. The generalized Cantor space 22 4. Generalized Borel sets 30 5. Generalized Borel functions 37 6. The generalized Baire space and Baire category 41 7. Standard Borel κ-spaces, κ-analyticquasi-orders,andspacesofcodes 47 8. Infinitary logics and models 55 9. κ-Souslin sets 65 10. The main construction 76 11. Completeness 85 12. Invariant universality 91 13. An alternative approach 106 14. Definable cardinality and reducibility 115 15. Some applications 126 16. Further completeness results 132 Indexes 147 Concepts 147 Symbols 148 Bibliography 151 iii Abstract We provide analogues of the results from [FMR11, CMMR13] (which correspond to the case κ = ω) for arbitrary κ-Souslin quasi-orders on any Polish space, for κ an infinite cardinal smaller than the cardinality of R. These generalizations yield a variety of results concerning the complexity of the embeddability relation between graphs or lattices of size κ, the isometric embeddability relation between complete metric spaces of density character κ, and the linear isometric embeddability relation between (real or complex) Banach spaces of density κ. Received by the editor March 19, 2019. 2010 Mathematics Subject Classification. 03E15, 03E60, 03E45, 03E10, 03E47.
    [Show full text]
  • Homogeneous Spaces and Wadge Theory
    Homogeneous spaces and Wadge theory Sandra M¨uller Universit¨atWien January 2019 joint work with Rapha¨elCarroy and Andrea Medini Arctic Set Theory Workshop 4 Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 1 How I got interested in general topology Our main tool: Wadge theory The beauty of Hausdorff operations Putting everything together Open questions and future goals Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 2 How I got interested in general topology Our main tool: Wadge theory The beauty of Hausdorff operations Putting everything together Open questions and future goals Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 3 X X h x y ! Examples of homogeneous spaces: all discrete spaces, Q, 2 , ! ! ≈ R n Q, all topological groups. We will focus on zero-dimensional homogeneous spaces, i.e. topological spaces which have a base of clopen sets. Fact X is a locally compact zero-dimensional homogeneous space iff X is discrete, X ≈ 2!, or X ≈ ! × 2!. We will therefore focus on non-locally compact (equivalently, nowhere compact) zero-dimensional homogeneous spaces. Homogeneous spaces All our topological spaces will be separable and metrizable. A homeomorphism between two spaces X and Y is a bijective continuous function f such that the inverse f −1 is continuous as well. Definition A space X is homogeneous if for every x; y 2 X there exists a homeomorphism h: X ! X such that h(x) = y. Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 4 ! Examples of homogeneous spaces: all discrete spaces, Q, 2 , ! ! ≈ R n Q, all topological groups.
    [Show full text]
  • Structural Dichotomy Theorems in Descriptive Set Theory
    Structural dichotomy theorems in descriptive set theory Benjamin Miller Universit¨atWien Winter Semester, 2019 Introduction The goal of these notes is to provide a succinct introduction to the primary structural dichotomy theorems of descriptive set theory. The only prerequisites are a rudimentary knowledge of point-set topology and set theory. Working in the base theory ZF + DC, we first discuss trees, the corresponding representations of closed, Borel, and Souslin sets, and Baire category. We then consider consequences of the open dihypergraph dichotomy and variants of the G0 dichotomy. While pri- marily focused upon Borel structures, we also note that minimal modi- fications of our arguments can be combined with well-known structural consequences of determinacy (which we take as a black box) to yield generalizations into the projective hierarchy and beyond. iii Contents Introduction iii Chapter 1. Preliminaries 1 1. Closed sets 1 2. Ranks 2 3. Borel sets 3 4. Souslin sets 4 5. Baire category 9 6. Canonical objects 13 Chapter 2. The box-open dihypergraph dichotomy 27 1. Colorings of box-open dihypergraphs 27 2. Partial compactifications 31 3. Separation by unions of closed hyperrectangles 34 Chapter 3. The G0 dichotomy, I: Abstract colorings 39 1. Colorings within cliques 39 2. Discrete perfect sets within cliques 42 3. Scrambled sets 45 Chapter 4. The G0 dichotomy, II: Borel colorings 49 1. Borel colorings 49 2. Index two subequivalence relations 52 3. Perfect antichains 54 4. Parametrization and uniformization 56 Chapter 5. The (G0; H0) dichotomy 61 1. Borel local colorings 61 2. Linearizability of quasi-orders 67 Bibliography 71 Index 73 v CHAPTER 1 Preliminaries 1.
    [Show full text]
  • Incompatible Ω-Complete Theories∗
    Incompatible Ω-Complete Theories∗ Peter Koellner and W. Hugh Woodin July 25, 2009 Abstract In 1985 the second author showed that if there is a proper class of mea- surable Woodin cardinals and V B1 and V B2 are generic extensions of V B1 B2 2 satisfying CH then V and V agree on all Σ1-statements. In terms of the strong logic Ω-logic this can be reformulated by saying that un- der the above large cardinal assumption ZFC + CH is Ω-complete for 2 2 Σ1. Moreover, CH is the unique Σ1-statement with this feature in the 2 sense that any other Σ1-statement with this feature is Ω-equivalent to CH over ZFC. It is natural to look for other strengthenings of ZFC that have an even greater degree of Ω-completeness. For example, one can ask for recursively enumerable axioms A such that relative to large cardinal axioms ZFC + A is Ω-complete for all of third-order arithmetic. Going further, for each specifiable segment Vλ of the uni- verse of sets (for example, one might take Vλ to be the least level that satisfies there is a proper class of huge cardinals), one can ask for recursively enumerable axioms A such that relative to large cardinal axioms ZFC + A is Ω-complete for the theory of Vλ. If such theories exist, extend one another, and are unique in the sense that any other such theory B with the same level of Ω-completeness as A is actually Ω-equivalent to A over ZFC, then this would show that there is a unique Ω-complete picture of the successive fragments of the universe of sets and it would make for a very strong case for axioms comple- menting large cardinal axioms.
    [Show full text]
  • AN Ω-LOGIC PRIMER Introduction One Family of Results in Modern Set
    AN ­-LOGIC PRIMER JOAN BAGARIA, NEUS CASTELLS, AND PAUL LARSON Abstract. In [12], Hugh Woodin introduced ­-logic, an approach to truth in the universe of sets inspired by recent work in large cardinals. Expository accounts of ­-logic appear in [13, 14, 1, 15, 16, 17]. In this paper we present proofs of some elementary facts about ­-logic, relative to the published literature, leading up to the generic invariance of ­- logic and the ­-conjecture. Introduction One family of results in modern set theory, called absoluteness results, shows that the existence of certain large cardinals implies that the truth values of certain sentences cannot be changed by forcing1. Another family of results shows that large cardinals imply that certain de¯nable sets of reals satisfy certain regularity properties, which in turn implies the existence of models satisfying other large cardinal properties. Results of the ¯rst type suggest a logic in which statements are said to be valid if they hold in every forcing extension. With some technical modi¯cations, this is Woodin's ­- logic, which ¯rst appeared in [12]. Results of the second type suggest that there should be a sort of internal characterization of validity in ­-logic. Woodin has proposed such a characterization, and the conjecture that it succeeds is called the ­-conjecture. Several expository papers on ­-logic and the ­-conjecture have been published [1, 13, 14, 15, 16, 17]. Here we briefly discuss the technical background of ­-logic, and prove some of the basic theorems in this area. This paper assumes a basic knowledge of Set Theory, including con- structibility and forcing.
    [Show full text]
  • These Are First-Draft Lecture Notes Issued for an Optional PG Theory
    DÖafØ ß ÒÓØ fÓÖ di×ØÖibÙØiÓÒ These are first-draft lecture notes issued for an optional PG theory course in Edinburgh. The course was concerned with descriptive hierarchies, specifically the arithmetical, analytic, and fixpoint hierarchies. The main aim was to go through the proof of the modal mu-calculus alternation hierarchy; on the way, a quick survey of the basics of effective descriptive set theory is given, together with a few of the more amusing set-theoretic facts. – JCB 1 Introduction Let us recall some basic facts about computable functions on the natural numbers, which were covered in earlier courses. We know that a (possibly partial) function is computable by a Turing machine iff it can be defined by means of a small set of operators, including in particular primitive recursion and minimalization. Accordingly we call a set of numbers ‘recursive’ if its characteristic function is computable. We know that there exist non-recursive sets and functions; indeed, we know that there are sets that are not even semi-recursive. (Recall that a set is semi-recursive if there is a program which confirms membership, even if it doesn’t necessarily terminate on a non-member. Equivalently, we can list all the members of the set; hence the equivalent term ‘recursively enumerable’, or r.e.) Given this, we might reasonably ask, what can we say about these ‘hard’ non-r.e. sets? How complicated can they be? And what is the right notion of complicated, anyway? One approach to answering these questions is to keep our focus on the notation of compu- tation, and start asking questions such as: if somebody gives me some non-recursive set as an oracle, what other sets can I then compute, using a standard Turing machine together with the oracle? If I then take a set I can’t even compute with my oracle, and then add that as an oracle, what happens? How far can I go on? Can I order sets in terms of the strength of the oracles needed to compute them? And so on.
    [Show full text]