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A New Finestructural Hierarchy for the Constructible Universe∗

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A New Finestructural Hierarchy for the Constructible Universe∗ A New Finestructural Hierarchy for the Constructible Universe¤ Peter Koepke Abstract: We present a natural hierarchy for G¨odel’smodel L of con- structible sets. The new hierarchy simplifies finestructural arguments. This is demonstrated by a proof of Jensen’s Covering Theorem for L. 1. Introduction The fundamental constructible operation is the formation of a definable subset of a level L® of G¨odel’sconstructible hierarchy: I(L®; '; ~y) = fx 2 L®jL® j= '[~y; x]g where ' is a first-order formula and ~y 2 L®. The next level of the hierarchy is then formed as L®+1 = fI(L®; '; ~y)j' is a formula; ~y 2 L®g: This hierarchy is analysed using Skolem functions S(L®; '; ~y) = the least x 2 L® such that L® j= '[~y; x] relative to a canonical wellorder <L® of L®. Whereas some principles like the Continuum Hypothesis can be proved in L with “coarse” methods, the proofs of subtle principles like Jensen’s 2 require a careful setup and analysis of these structures. In Jensen’s finestructure theory of the constructible hierarchy [4], first-order definability is split up into iterated Σ1-definability over appropriate structures. It is essential that ¤This work was carried out during research visits to the Centre de Recerca Mate- m`atica, Barcelona, Spain in 2001 and 2003. The author wishes to thank the CRM and its members for their generous support and hospitality. 1 the constructible operations are definable by 2-formulas of low complexity which has to be shown with considerable effort. This article presents an approach which aims at reducing these complica- tions: the new hierarchy is built with a definability notion of the order of Σ1-definability; the corresponding constructible operations and notions are incorporated into the language as basic symbols. Combinatorial applications of finestructure theory are based on the uniform behaviour of hulls under finestructural Skolem functions. Constellations in the constructible universe can be captured by finestructural hulls; the condensation lemma controls the shape of such hulls since they have to be isomorphic to levels of the hierarchy. The hulls formed in the new finestructure satisfy laws known from the clas- sical theory, and they are therefore adequate for finestructural arguments. We demonstrate this by a proof of the Jensen Covering Theorem for L which is simpler than the original proof in [1] since it need not distinguish cases according to the various complexities of definitions. The article is structured as follows: Sections 2 - 5 introduce the basic no- tions of the fine hierarchy; we prove condensation and an analogous result about directed limits. Section 6 transfers Jensen’s upward extensions of embeddings technique to the fine hierarchy; the method basically defines an ultrapower by an extender, so we call the resulting map an extension. In section 7 we construct strong maps with wellfounded extensions. The tech- niques are put together in section 8 for a proof of the Covering Theorem. The techniques of the proof of the Covering Theorem can be used in core model theory. In particular, fine ultrapowers and extensions can be formed by the techniques of section 6. 2. The Fine Hierarchy We approximate the constructible universe by a hierarchy (F®)®2On of structuresS F® = (F®;I ¹ F®;S ¹ F®; 2; <¹ F®). Each F® is a transitive set and ®2On F® = L. The functions I ¹ F® and S ¹ F® are the restrictions to <! F® of a global interpretation function I and a global Skolem function S <! defined on L . The relation <¹ F® is the restriction of a ternary guarded 3 constructible wellorder to F®. We shall usually write (F®; I; S; 2; <) instead of F® = (F®;I ¹ F®;S ¹ F®; 2; <¹ F®). We first define a language adequate for the structures F®. 2.1. Definition. Let L be the first-order language with the following components: 2 ² variable symbolsv ˙n for n < !; ² logical symbols ˙=(equality), ^˙ (conjunction),: ˙ (negation), 9˙ (existential quantification), (, ) (brackets); ² function symbols I˙ (interpretation), S˙ (Skolem function) of variable finite arity; ² a binary relation symbol 2˙ (set-membership) and a ternary relation sym- bol <˙ (guarded wellorder). The syntax and semantics of L are defined as usual. The variable arity of I˙ and S˙ is handled by bracketing: if n < ! and t0; : : : ; tn¡1 are L-terms then I˙(t0; : : : ; tn¡1) and S˙(t0; : : : ; tn¡1) are L-terms. If t0, t1, t2 are L-terms then ˙ ˙ t02t1 and t0<t1 t2 are atomic L-formulas. We denote the set of first-order L-formulas simply by L. We assume that L is G¨odelisedconveniently: the set of G¨odelisedformulas satisfies L ⊆ fVn j n < !g (this technical condition will be used in used in Proposition 3.6); the usual syntactical operations of L are recursively defin- ~t ~ able over V!. This includes the simultaneous substitution ' ~w of terms t for variables ~w in '. The notation '(v ˙0;:::; v˙n¡1) implies that the free vari- ables of ' are contained in fv˙0;:::; v˙n¡1g. By L0 we denote the collection of quantifier-free formulas of L. The language L is interpreted in L-structures in the obvious way. If A = (A; : : : ) is an L-structure, '(v ˙0;:::; v˙n¡1) 2 L and a0; : : : ; an¡1 2 A then A j= '[a0; : : : ; an¡1] means that A is a model of ' under the variable assignmentv ˙i 7! ai for i < n. The F®-hierarchy is defined by iterated L0-definability: 2.2. Definition. The fine hierarchy consists of fine levels F® = (F®;I ¹ F®;S ¹ F®; 2; <¹ F®) which are defined by recursion on ® 2 On: For ® · ! let F® = V® and 8~x 2 F® : I(~x) = S(~x) = 0. Let <F! be a binary relation which wellorders F! = V! in ordertype ! and which extends the 2-relation. Define the ternary relation <¹ F! by: x <y z iff (y = Fn for some n < !, x; z 2 y and x <F! z). This defines the structures F® for ® · !. Assume that ® ¸ ! and that F® = (F®;I ¹ F®;S ¹ F®; 2; <¹ F®) has been defined. For '(v ˙0;:::; v˙n) 2 L0 and x0; : : : ; xn¡1 2 F® set (¤) I(F®; '; x0; : : : ; xn¡1) = fxn 2 F® j F® j= '[x0; : : : ; xn]g. We say that (F®; '; x0; : : : ; xn¡1) is a name for its interpretation I(F®; '; x0; : : : ; xn¡1) in the fine hierarchy. 3 The next fine level is defined as F®+1 = fI(F®; '; x0 : : : xn¡1) j '(v ˙0 ::: v˙n) 2 L0; x0 : : : xn¡1 2 F®; n < !g: To define I ¹ F®+1 we only need to define I(~z) for “new” vectors ~z 2 <! <! (F®+1) n F® . We have already made certain assignments in (¤); in all other cases set I(~z) = 0. Define the relation <F® endextending <F! : for x; y 2 F® n F! set x <F® y iff there is a name (F¯; '; x0; : : : ; xm¡1) for x such that every name (Fγ ; Ã; y0; : : : ; yn¡1) for y is lexicographically greater than (F¯; '; x0; : : : ; xm¡1), where the first two coordinates are wellordered by 2 and the further coordinates are wellordered by <Fγ . The ternary re- lation <¹ F®+1 is then defined by: x <y z iff y = Fº for some º · ® and x <Fº z. The Skolem function S finds witnesses of existential state- <! <! ments. Again we only need to define S(~z) for ~z 2 (F®+1) n F® . We set S(~z) = 0 except when ~z = F®;'(v ˙0;:::; v˙n); x0; : : : ; xm¡1, where ' 2 L0; x0; : : : ; xm¡1 2 F®, m · n, and there exist xm; xm+1; : : : ; xn 2 F® such that: F® j= '[x0; : : : ; xm¡1; xm; xm+1; : : : ; xn]; in this case let S(~z) be such an xm minimal with respect to the wellfounded relation <F® . This defines F®+1 = (F®+1;I ¹ F®+1;S ¹ F®+1; 2; <¹ F®+1). Assume thatS ¸ > ! is a limit ordinal and that F® is defined for ® < ¸. Then let F¸ = ®<¸ F®, which determines F¸ = (F¸;I ¹ F¸;S ¹ F¸; 2; <¹ F¸). The fine hierarchy satisfies basic hierarchical properties some of which were assumed tacitely in the previous definition: 2.3. Proposition. For every γ 2 On: (a) ® · γ ! F® ⊆ Fγ ; (b) ® < γ ! F® 2 Fγ ; (c) Fγ is transitive. Proof. By simultaneous induction on γ. Assume that (a) – (c) hold for ¯ < γ. Then they trivially hold at γ if γ · ! or γ is a limit ordinal. Consider the remaining case γ = ¯ + 1, ¯ ¸ !. (a) It suffices to show that F¯ ⊆ F¯+1. Let z 2 F¯. Since F¯ is transitive, z = fx 2 F¯ j x 2 zg = fx 2 F¯ j F¯ j= (v ˙12˙ v˙0)[z; x]g 2 F¯+1. (b) By (a), it suffices to show that F¯ = fx 2 F¯ j F¯ j= (v ˙0 ˙=˙v0)[x]g 2 F¯+1. (c) Let a 2 F¯+1. Then a ⊆ F¯ ⊆ F¯+1, and a ⊆ F¯+1. qed First-order definability can be emulated in S0: 4 2.4. Proposition. For every 2-formula '(v ˙0;:::; v˙m¡1) one can uniformly ¤ define a quantifier-free formula ' (v ˙0;:::; v˙m¡1; v˙m;:::; v˙m+k¡1) 2 L0 such that for all ® ¸ ! and for all a0; : : : ; am¡1 2 F®: ¤ (F®; 2)j='[a0; : : : ; am¡1] iff F®+kj=' [a0; : : : ; am¡1;F®;:::;F®+(k¡1)]: Proof. By induction on the complexity of '. For atomic ' let '¤ = ', ¤ ¤ ¤ ¤ ¤ for ' =: ˙ '0 let ' =: ˙ '0, and for ' = '0^˙ '1 set ' = '0^˙ '1. The only nontrivial case is the existential one. Consider '; '¤ satisfying the ˙ proposition. Then: (F®; 2) j= 9v˙j'[a0; : : : ; am¡1] () 9bj 2 F® (F®; 2) j= '[a0; : : : ; aj¡1; bj; aj+1; : : : ; am¡1] ¤ () 9bj 2 F® F®+k j= ' [a0; : : : ; bj; : : : ; am¡1;F®;:::;F®+(k¡1)] ¤ v˙m+k () fxm+k 2 F® j F®+k j= (' )[a0; : : : ; am¡1;F®;:::; v˙j F®+(k¡1); xm+k]g 6= ; ¤ v˙m+k () fxm+k 2 F®+k j F®+k j= (v ˙m+k2˙ v˙m^˙ ' ) v˙j [a0;:::;F®; : : : ; xm+k]g 6= ; ¤ v˙m+k () I(F®+k; (v ˙m+k2˙ v˙m^˙ ' ); a0; : : : ; am¡1;F®;:::;F ) 6= ;.
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