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 .