DOCSLIB.ORG

Simplified Constructibility Theory∗

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Simplified Constructibility Theory∗ Simplified Constructibility Theory∗ Minicourse Helsinki, March 2005 by Peter Koepke University of Bonn, Germany Web: http: //www. math. uni -bonn. de/people/logi c /People/Koepke. html . This document has been written using the GNU T X text editor ( see www. texmacs . org) . ∗ E MAC S Preface This manuscript covers the contents of a minicourse held at the University of Helsinki, Finland in March 2005. We made a fast four-day journey through con- structibility theory, starting from first motivations and reaching the heights of morass constructions. This was possible on the basis of hyperfine structure theory - a simplified approach to the fine structure theory of the constructible universe. Still, we had to concentrate on the description of fundamental ideas; only few arguments could be carried out completely. This text fills in some more details which are characteristic for the hyperfine theory. It should enable the reader to complete the proofs by himself. Hyperfine structure theory was introduced by S y D. friedman and the pre- sent author [ 4] . The standard reference on constructibility theory was written by Keith J. Devlin [ 3] . Suggestions for further reading on constructibility theory and simplified fine structure are contained in the bibliography. I am grateful for a mathematically rich and very enjoyable fortnight with the Helsinki logic group, who showed a strong interest in the subject of the course. My special thanks go to Prof. Jouko V¨a¨an¨anen and Prof. Juliette Kennedy for their support and warm hospitality. Peter Koepke Table of contents Table of contents . 3 1 The continuum problem . 4 ¨ 2 ¢¡¤£¤¥§¦ ' s relative consistency results . 4 3 The constructible hierarchy . 5 4 Wellordering L . 8 5 An external continuum hypothesis . 9 6 The axiom of constructibility . 1 0 7 C ondensation . 1 0 8 C onstructible operations . 1 1 9 The generalized continuum hypothesis in L . 1 4 1 0 Finestructure . 1 5 1 1 The fine hierarchy . 1 6 1 2 Fine hulls . 1 7 1 3 Definition of a gap-1 morass . 20 1 4 Proving the morass axioms . 23 Bibliography . 29 4 S ection 2 1 The continuum problem The creation of set theory by Georg Cantor was initiated and guided by his study of sets of real numbers. In 1 873 he showed that the set of reals is uncountable [ 1 ] . He defined that two sets X ; Y have the same cardinality, card( X) = card( Y) or X Y, if there is a bijection f: X Y. Thus ∼ $ ¡ ¡ , i. e. , f f: : :9 $ Further research on the cardinality of and its subsets lead Cantor to the for- mulation of the continuum hypothesis ( CH) in 1 878 [ 2] : card( ) = ; @ 1 where 0; 1 ; ¢ is the increasing sequence of infinite cardinals. The continuum hypoth@esis @is equivalent to X ( ( n !X n) X ) : 8 ⊆ 9 6 ∼ _ ∼ In cardinal arithmetic this is usually written as 2 @ 0 = ; @ 1 κ ¡ where 2 = card( ( κ) ) and is identified with the powerset of . P In 1 908 Felix Hausdorff [ 1 0] extended this to the generalized continuum hypothesis ( GCH) : α 2 @ α = : 8 @ α + 1 Contrary to the expectations of Cantor, David Hilbert and other mathe- maticians these hypotheses could not be proved or decided on the basis of the usual set theoretic assumptions and intuitions. To examine unprovability , the underlying axioms and logical rules have to be specified; we shall work in Z er- melo-Fraenkel set theory ZF and usual first-order predicate logic. The system ZF is formulated in the language whose only non-logical symbol is the binary -relation. f 2 g 2 2 £¥¤§¦©¨ ' s relative consistency results Kurt Go¨ del proved the unprovability of the negation of the continuum hypoth- esis, i. e. , its ( relative) consistency, in notes and articles published between 1 938 and 1 940 [ 6] , [ 8] , [ 7] , [ 9] . He presents his results in various forms which we can subsume as follows: there is an -term L such that 2 ZF ` ` ( L ; ) ZF + the axiom of choice ( AC) + GCH" : ` 2 So ZF sees a model for the stronger theory ZF + AC + GCH. If the system ZF is consistent, then so is ZF+AC+GCH. In ZF, the term L has a host of special properties; L is the -minimal inner model of ZF, i. e. , the -smallest model of ZF which is transitive⊆ and contains the class Ord of ordinals. ⊆ The constructible hierarchy 5 The construction of L is motivated by the idea of recursively constructing a minimal model of ZF. The archetypical ZF-axiom is Zermelo' s comprehension schema ( axiom of subsets ) : for every -formula '( v ; w ) postulate 2 x p v x '( v; p ) V: 8 8 f 2 j g 2 The term V denotes the abstraction term v v = v , i. e. , the set theoretic uni- verse ; formulas with abstraction terms arefabj breviagtions for pure -formulas. E. g. , the above instance of the comprehension schema abbreviates the 2formula x p y v ( v y v x '( v ; p ) ) : 8 8 9 8 2 $ 2 ^ The basic idea for building a ( minimal) model of set theory would be to form a kind of closure under the operations ¡ ( x ; p ) v x '( v; p ) : f 2 j g However, there is the difficulty where to evaluate the formula '. The comprehen- sion instance should be satisfied in the model to be built, i. e. , the quantifiers of ' may have to range about sets which have not yet been included in the construc- tion. To avoid this the evaluation of the formula will only refer to sets already constructed and we shall consider the modified definability operations ¡ ( x; p ) v x ( x ; ) '( v; p ) : f 2 j 2 g This could be termed a predicative operation whereas the strong operation would be an impredicative one. The set v x ( x ; ) '( v; p ) is determined by fixing f 2 j 2 g x ; ' ; p . One can thus view v x ( x ; ) '( v ; p ) as an interpretation of the f 2 j 2 g name ( x; ' ; p ) . These ideas will be used in the definition of the constructible hierarchy . 3 The constructible hierarchy Unlike the impredicative definability operation the predicative operation can be defined as a single set theoretic operation in ZF. We assume that the -formulas are g¨odelized by natural numbers in a recursive way so that every form2ula is < - larger than its proper subformulas; let Fml be the set of -formulas in that G¨odelization. We may assume that 0 is the smallest element o2f Fml. Then define the satisfaction predicate ( x ; ) '( a ) 2 for ( x; ' ; a ) V Fml V and the interpretation operation 2 × × I( x ; ' ; a ) = v x ( x; ) '( v ; a ) : f 2 j 2 g One can view Def( x) = I( x; ' ; p ) ' Fml; p x f j 2 2 g as a definable or predicative powerset of x. The constructible hierarchy is obtained by iterating the Def-operation along the ordinals. 6 S ection 3 Definition 1 . Define the constructible hierarchy Lα , α Ord by recursion on α: 2 L = 0 ; L α+ 1 = Def( L α) L λ = Lα , for λ a limit ordinal : α < λ [ The constructible universe L is the union of that hierarchy: L = Lα : α Ord 2[ The hierarchy satisfies natural hierarchical laws. Theorem 2. a ) α β implies L L 6 α ⊆ β b ) L β is transitive c ) L V β ⊆ β d ) α < β implies L L α 2 β e ) L Ord = β β \ f) β 6 ! implies L β = Vβ g) β > ! implies card( L β) = card( β) Proof. By induction on β Ord. The cases β = 0 and β a limit ordinal are easy 2 and do not depend on the specific definition of the L β-hierarchy. Let β = γ + 1 where the claims hold for γ. a) It suffices to show that L γ L β . Let x L γ . By b) , L γ is transitive and x L . Hence ⊆ 2 ⊆ γ x x = v L v x = v L ( L ; ) ( v w) = I( L ; v w ; x) L = L : f 2 γ j 2 g f 2 γ j γ 2 2 w g γ 2 2 γ+ 1 β b) Let x L β . Let x = I( L γ; ' ; p ) . Then by a) x L γ L β . c) By ind2uction hypothesis, ⊆ ⊆ L = Def( L ) ( L ) ( V ) = V = V : β γ ⊆ P γ ⊆ P γ γ+ 1 β d) It suffices to show that L L . γ 2 β L = v L v = v = v L ( L ; ) v = v = I( L ; v = v ; ) L = L : γ f 2 γ j g f 2 γ j γ 2 g γ ; 2 γ+ 1 β e) L Ord V Ord = β. For the converse, let δ < β. If δ < γ the inductive β \ ⊆ β \ hypothesis yields that δ L γ Ord L β Ord. Consider the case δ = γ. We have to show that γ L . Th2ere is\ a form⊆ula \'( v) which is Σ and formalizes being an 2 β 0 ordinal. This means that all quantifiers in ' are bounded and if z is transitive then v z ( v Ord ( z ; ) '( v) ) : 8 2 2 $ 2 The constructible hierarchy 7 By induction hypothesis γ = v L v Ord f 2 γ j 2 g = v L ( L ; ) '( v) f 2 γ j γ 2 g = I( L ; ' ; ) γ ; L = L : 2 γ+ 1 β f) Let β < !. By c) it suffices to see that V L . Let x V . By induction β ⊆ β 2 β hypothesis, L γ = Vγ . x Vγ = L γ . Let x = x0; ¢ ; xn 1 .
Load more

Recommended publications
  • Set-Theoretic Geology, the Ultimate Inner Model, and New Axioms
    Set-theoretic Geology, the Ultimate Inner Model, and New Axioms Justin William Henry Cavitt (860) 949-5686 [email protected] Advisor: W. Hugh Woodin Harvard University March 20, 2017 Submitted in partial fulfillment of the requirements for the degree of Bachelor of Arts in Mathematics and Philosophy Contents 1 Introduction 2 1.1 Author’s Note . .4 1.2 Acknowledgements . .4 2 The Independence Problem 5 2.1 Gödelian Independence and Consistency Strength . .5 2.2 Forcing and Natural Independence . .7 2.2.1 Basics of Forcing . .8 2.2.2 Forcing Facts . 11 2.2.3 The Space of All Forcing Extensions: The Generic Multiverse 15 2.3 Recap . 16 3 Approaches to New Axioms 17 3.1 Large Cardinals . 17 3.2 Inner Model Theory . 25 3.2.1 Basic Facts . 26 3.2.2 The Constructible Universe . 30 3.2.3 Other Inner Models . 35 3.2.4 Relative Constructibility . 38 3.3 Recap . 39 4 Ultimate L 40 4.1 The Axiom V = Ultimate L ..................... 41 4.2 Central Features of Ultimate L .................... 42 4.3 Further Philosophical Considerations . 47 4.4 Recap . 51 1 5 Set-theoretic Geology 52 5.1 Preliminaries . 52 5.2 The Downward Directed Grounds Hypothesis . 54 5.2.1 Bukovský’s Theorem . 54 5.2.2 The Main Argument . 61 5.3 Main Results . 65 5.4 Recap . 74 6 Conclusion 74 7 Appendix 75 7.1 Notation . 75 7.2 The ZFC Axioms . 76 7.3 The Ordinals . 77 7.4 The Universe of Sets . 77 7.5 Transitive Models and Absoluteness .
    [Show full text]
  • Recognizable Sets and Woodin Cardinals: Computation Beyond the Constructible Universe
    RECOGNIZABLE SETS AND WOODIN CARDINALS: COMPUTATION BEYOND THE CONSTRUCTIBLE UNIVERSE MERLIN CARL, PHILIPP SCHLICHT AND PHILIP WELCH Abstract. We call a subset of an ordinal λ recognizable if it is the unique subset x of λ for which some Turing machine with ordinal time and tape, which halts for all subsets of λ as input, halts with the final state 0. Equivalently, such a set is the unique subset x which satisfies a given Σ1 formula in L[x]. We prove several results about sets of ordinals recognizable from ordinal parameters by ordinal time Turing machines. Notably we show the following results from large cardinals. • Computable sets are elements of L, while recognizable objects appear up to the level of Woodin cardinals. • A subset of a countable ordinal λ is in the recognizable closure for subsets of λ if and only if it is an element of M 1, 1 where M denotes the inner model obtained by iterating the least measure of M1 through the ordinals, and where the recognizable closure for subsets of λ is defined by closing under relative recognizability for subsets of λ. Contents 1. Introduction1 2. Definitions and basic facts3 3. Recognizable sets5 3.1. Machines with fixed ordinal parameters6 3.2. Different ordinals9 3.3. Arbitrary ordinals 10 3.4. The recognizable closure and HOD 13 4. Recognizable sets and M1 14 4.1. Subsets of countable ordinals 14 4.2. Subsets of !1 17 4.3. Structures which are not recognizable 18 4.4. Generically recognizable sets 20 5. Conclusion and questions 20 References 21 1.
    [Show full text]
  • Ronald Jensen Receives a 2003 Steele Prize Gottfried Wilhelm-Leibniz-Preisträger 2003
    Allyn Jackson Ronald Jensen receives a 2003 Steele Prize At the 109th Annual Meeting of the AMS in Baltimore in January 2003, a Steele Prize has been awarded to Ronald Jensen for a Seminal Contribution to Research. The text that follows presents the selection committee’s citation and a brief biographical sketch. (Allyn Jackson) Ronald Jensen’s paper,“The fine structure of the con- of an inner model, in this case G¨odel’s constructible structible hierarchy” (Annals of Mathematical Logic sets, that go far beyond G¨odel’s proof of the general- 4 (1972) 229–308), has been of seminal importance ized continuum hypothesis in this model. for two different directions of research in contempo- The second direction initiated by Jensen’s paper rary set theory: the inner model program and the use involves applying these combinatorial principles to of combinatorial principles of the sort that Jensen es- problems arising in other parts of mathematics. The tablished for the constructible universe. principle, which Jensen proved to hold in the con- The inner model program, one of the most active structible universe, has been particularly useful in parts of set theory nowadays, has as its goals the un- such applications. By now, it has become part of the derstanding of very large cardinals and their use to standard tool kit of several branches of mathematics, measure the consistency strength of assertions about ranging from general topology to module theory. much smaller sets. A central ingredient of this pro- Ronald Jensen received his Ph.D. in 1964 from the gram is to build, for a given large cardinal axiom, a University of Bonn.
    [Show full text]
  • The Semi-Weak Square Principle
    The Semi-Weak Square Principle Maxwell Levine Universit¨atWien Kurt G¨odelResearch Center for Mathematical Logic W¨ahringerStraße 25 1090 Wien Austria [email protected] Abstract Cummings, Foreman, and Magidor proved that for any λ < κ, κ,λ implies that ∗ κ carries a PCF-theoretic object called a very good scale, but that κ (which one could write as κ,κ) is consistent with the absence of a very good scale at κ. They asked whether κ,<κ is enough to imply the existence of a very good scale for κ, and we resolve this question in the negative. Furthermore, we sharpen a theorem of Cummings and Schimmerling and show that κ,<κ implies the failure of simultaneous stationary reflection at κ+ for any singular κ. This implies as a corollary that if Martin's Maximum holds, then κ,<κ fails for singular cardinals κ of cofinality !1. Keywords: Set Theory, Forcing, Large Cardinals 1. Background Singular cardinals occupy a curious place in set theory. They are amenable to the forcing technique devised by Cohen to establish the independence of the continuum hypothesis, but many of their properties can be proved outright from ZFC using the PCF theory developed by Shelah. The properties of singular cardinals in a given model depend on a fundamental tension between the extent to which the model supports large cardinals, and the extent to which it resembles G¨odel'sconstructible universe L. The square principle, denoted κ if it holds at a cardinal κ, was distilled by Jensen in order to find Suslin trees at successors of singular cardinals in L [8], and it embodies many of the constructions that can be carried out in L.
    [Show full text]
  • Forcing? Thomas Jech
    WHAT IS... ? Forcing? Thomas Jech What is forcing? Forcing is a remarkably powerful case that there exists no proof of the conjecture technique for the construction of models of set and no proof of its negation. theory. It was invented in 1963 by Paul Cohen1, To make this vague discussion more precise we who used it to prove the independence of the will first elaborate on the concepts of theorem and Continuum Hypothesis. He constructed a model proof. of set theory in which the Continuum Hypothesis What are theorems and proofs? It is a use- (CH) fails, thus showing that CH is not provable ful fact that every mathematical statement can from the axioms of set theory. be expressed in the language of set theory. All What is the Continuum Hypothesis? In 1873 mathematical objects can be regarded as sets, and Georg Cantor proved that the continuum is un- relations between them can be reduced to expres- countable: that there exists no mapping of the set sions that use only the relation ∈. It is not essential N of all integers onto the set R of all real numbers. how it is done, but it can be done: For instance, Since R contains N, we have 2ℵ0 > ℵ , where 2ℵ0 0 integers are certain finite sets, rational numbers and ℵ are the cardinalities of R and N, respec- 0 are pairs of integers, real numbers are identified tively. A question arises whether 2ℵ0 is equal to with Dedekind cuts in the rationals, functions the cardinal ℵ1, the immediate successor of ℵ0.
    [Show full text]
  • Inner Models from Extended Logics: Part 1*
    Inner Models from Extended Logics: Part 1* Juliette Kennedy† Menachem Magidor‡ Helsinki Jerusalem Jouko Va¨an¨ anen¨ § Helsinki and Amsterdam December 15, 2020 Abstract If we replace first order logic by second order logic in the original def- inition of Godel’s¨ inner model L, we obtain HOD ([33]). In this paper we consider inner models that arise if we replace first order logic by a logic that has some, but not all, of the strength of second order logic. Typical examples are the extensions of first order logic by generalized quantifiers, such as the Magidor-Malitz quantifier ([24]), the cofinality quantifier ([35]), or station- ary logic ([6]). Our first set of results show that both L and HOD manifest some amount of formalism freeness in the sense that they are not very sen- sitive to the choice of the underlying logic. Our second set of results shows that the cofinality quantifier gives rise to a new robust inner model between L and HOD. We show, among other things, that assuming a proper class of Woodin cardinals the regular cardinals > @1 of V are weakly compact in the inner model arising from the cofinality quantifier and the theory of that model is (set) forcing absolute and independent of the cofinality in question. *The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for its hospitality during the programme Mathematical, Foundational and Computational Aspects of the Higher Infinite supported by EPSRC Grant Number EP/K032208/1. The authors are grateful to John Steel, Philip Welch and Hugh Woodin for comments on the results presented here.
    [Show full text]
  • Set Theory in Second-Order
    Set theory in second-order Victoria Gitman [email protected] http://boolesrings.org/victoriagitman Set Theory in UK 2 May 8, 2019 Victoria Gitman Second-order set theory STUK2 1 / 31 Classes in naive set theory The absence of a formal distinction between sets and classes in naive set theory led to supposed paradoxes. Russell's Paradox Let X be the set of all sets that are not members of themselves. Assuming X 2 X implies X 2= X and assuming X 2= X implies X 2 X . Burali-Forti's Paradox The set of all ordinals is itself an ordinal larger than all ordinals and therefore cannot be an ordinal. Victoria Gitman Second-order set theory STUK2 2 / 31 Classes in first-order set theory The objects of first-order set theory are sets. Definition:A class is a first-order definable (with parameters) collection of sets. Classes play an important role in modern set theory. Inner models Ord-length products and iterations of forcing notions Elementary embeddings of the universe But first-order set theory does not provide a framework for understanding the structure of classes. In first-order set theory we cannot study: non-definable collections of sets, properties which quantify over classes. Second-order set theory is a formal framework in which both sets and classes are objects of the set-theoretic universe. In this framework, we can study: non-definable classes general properties of classes Victoria Gitman Second-order set theory STUK2 3 / 31 Why second-order set theory? Kunen's Inconsistency How do we formalize the statement of Kunen's Inconsistency that there is no non-trivial elementary embedding j : V ! V ? The following result is nearly trivial to prove.
    [Show full text]
  • Chapter 2 Constructible Sets
    Chapter 2 Constructible sets In this chapter, we recall Gödel’s construction of the constructive universe L and prove that both the Axiom of Choice (AC) and the Generalized Continuum Hypothesis (GCH) hold in this universe. Throughout this chapter, we informally work in ZF unless otherwise stated. 2.1 The set of formulas In Sections 1.1.1 and 1.2.1, we defined the language of set theory at the meta-theoretic level. Up to now, formulas of ZF are syntactic entities that are external to set theory. However, Gödel’s construction relies on the central concept of definability (Section 2.2), that requires to manipulate formulas within the set-theoretic universe, the same way as we manipulate sets. For that, we now need to internalize the language of formulas of ZF in ZF, by constructing a particular denumerable set Form whose elements—called internal formulas or codes of formulas—are intended to represent external formulas as sets. 2.1.1 Construction of the set Form The construction of the set Form closely follows the inductive definition of the syntactic cate- gory of formulas in the meta-language (Sections 1.1.1 and 1.2.1). Variables are represented by natural numbers, letting Var = ω. When they are used as (internal) variables, natural numbers are written using the typewriter letters x, y, z, etc. (to prevent confusions with external variables, that are still written using italic letters x, y, z, etc.) For simplicity, we only consider formulas that are constructed from equality, membership, negation, disjunction and existential quantification, since all the other constructions of first- order logic can be defined from the latter using De Morgan laws.
    [Show full text]
  • How the Continuum Hypothesis Can Be Solved (Or Not)
    True, false, independent: how the Continuum Hypothesis can be solved (or not). Carolin Antos 12.12.2017 Zukunftskolleg University of Konstanz 1 Structure of the talk 1. The Continuum Hypothesis 2. The constructible universe L 3. Forcing 4. Outlook: CH and the multiverse 2 The Continuum Hypothesis Infinite cardinalities The Continuum Hypothesis (CH), Cantor, 1878 There is no set whose cardinality is strictly between that of the @ natural and the real numbers: j P(N) j = 2 0 = @1. • Question arises from Cantor's work on ordinals and cardinals: j N j = @0, but what is j R j ? • Cantor tried to prove the CH but did not succeed. • Hilbert posed the CH as the first problem on his list of important open questions in 1900. 3 Independence Incompleteness Theorem, G¨odel,1931 Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F . • A statement that cannot be proved or disproved from such a system F is called independent from F . • Independence is important for finding axioms. • But: No matter how many axioms one adds, the system will never be complete. 4 Independence from ZFC? Standard axiomatization of set theory ZFC: • Extensionality. • Pairing. • Union. • Infinity. • Power Set. • Foundation. • Replacement. • Comprehension. • Choice. 5 Independence from ZFC? To show that CH is independent from ZF(C) we have to show that: 1. CH can be added to ZF(C) as an axiom and the resulting theory is consistent iff ZF(C) is consistent, and 2.
    [Show full text]
  • Representing Sets of Ordinals As Countable Unions of Sets in the Core Model
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 317, Number I, January 1990 REPRESENTING SETS OF ORDINALS AS COUNTABLE UNIONS OF SETS IN THE CORE MODEL MENACHEM MAGIDOR ABSTRACT. We prove the following theorems. Theorem I (~O#). Every set of ordinals which is closed under primitive recursive set functions is a countable union of sets in L. Theorem 2. (No inner model with an Erdos cardinal, i.e. K -> (wtJ<w.) For every ordinal f1, there is in K an algebra on f1 with countably many opera- tions such that every subset of f1 closed under the operations of the algebra is a countable union of sets in K. O. INTRODUCTION (See [Je2] for basic notation and terminology.) Jensen's famous covering theroem for L [De-Jen] claims that if 0# does not exist then every set of ordinals X is included in a set of ordinals Y E L such that IYI :::; IXI + N) . In some sense the meaning of this theorem is that if 0# does not exist, then the universe of sets V is not too far from the constructible universe L. We can consider the set Y in the theorem to be an approximation for X from above. This paper deals with the problem of "approximating the set X by a set in L from below". For instance consider the following problem: Assume 0# does not exist (-,0#) and X is uncountable. Does X necessarily contain an uncountable subset which is in L? The answer to this problem as stated is obviously "No". Consider a subset of N) , introduced by forcing over L, using finite conditions.
    [Show full text]
  • SETS of ORDINALS CONSTRUCTIBLE from TREES and the THIRD VICTORIA DELFINO PROBLEM Howards. Becker(!) and Alexander S. Kechris(Z)
    Contemporary Mathematics Volume 31, 1984 SETS OF ORDINALS CONSTRUCTIBLE FROM TREES AND THE THIRD VICTORIA DELFINO PROBLEM HowardS. Becker(!) and Alexander S. Kechris(z) A very important part of the structure theory of z~ sets of reals is based on their close interrelationship with the GBdel constructible universe L. The fundamental fact underlying this connection is the theorem of Shoenfield which asserts that every z1 set of reals is Souslin over L. This means that given any z1 subset of the reals (= ww in this paper), there is a tree T on w x A (A some ordinal, which can be taken to be ~l here) such that TeL and A= p[T] = {a € ww: :iff € Aw vn(arn, trn) € T}. In 1971 Moschovakis [14] introduced the notion of scale and showed that under the hypothesis of Projective Determinacy (PD) the pointclasses TI~n+l' l z2n+2 for all n .:._ 0 have the Sea 1e Property. To take for instance the case of TI~n+l this asserts that given any TI~n+l set As ww there is a sequence of norms ~m= A+ Ordinals (which we are always assuming here to be regular, i.e., having as range an initial segment of the ordinals) such that the follow­ ing properties hold: (i) If a0,a1, ... eA, ai+a and ~m(ai)+Am (inthesensethat '.Pm(ai) = Am for all large enough i), then a e A and ~m(a) 2_ Am. (ii) The relations (on m, a, S): a 2_~* 8 ~a € A 1\ (S ~ Avq:>m(o:) 2.~m{S)), * m 1 a:< S ~a € A 1\ (S ~A v ~(a)<~ (S)) are TI n+l' Cl'm m m 2 Such a sequence {cpm} is called a (regular) TI~n+ 1 -saale on A.
    [Show full text]
  • The Continuum Hypothesis, the Generic-Multiverse of Sets, and the Ω Conjecture
    The Continuum Hypothesis, the generic-multiverse of sets, and the Ω Conjecture W. Hugh Woodin July 16, 2009 1 A tale of two problems The formal independence of Cantor’s Continuum Hypothesis from the axioms of Set Theory (ZFC) is an immediate corollary of the following two theorems where the state- ment of the Cohen’s theorem is recast in the more modern formulation of the Boolean valued universe. Theorem 1 (Godel,¨ [3]). Assume V = L. Then the Continuum Hypothesis holds. ut Theorem 2 (Cohen, [1]). There exists a complete Boolean algebra, B, such that VB “The Continuum Hypothesis is false”: ut Is this really evidence (as is often cited) that the Continuum Hypothesis has no answer? Another prominent problem from the early 20th century concerns the projective sets, [8]; these are the subsets of Rn which are generated from the closed sets in finitely many steps taking images by continuous functions, f : Rn ! Rn, and complements. A function, f : R ! R, is projective if the graph of f is a projective subset of R × R. Let Projective Uniformization be the assertion: For each projective set A ⊂ R × R there exists a projective function, f : R ! R, such that for all x 2 R if there exists y 2 R such that (x; y) 2 A then (x; f (x)) 2 A. The two theorems above concerning the Continuum Hypothesis have versions for Projective Uniformization. Curiously the Boolean algebra for Cohen’s theorem is the same in both cases, but in case of the problem of Projective Uniformization an addi- tional hypothesis on V is necessary.
    [Show full text]