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AN Ω-LOGIC PRIMER Introduction One Family of Results in Modern Set

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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. All unde¯ned notions can be found in [4]. 1. ²­ 1.1. Preliminaries. Key words and phrases. ­-logic { Woodin cardinals { A-closed sets { universally Baire sets { ­-conjecture. The ¯rst author was partially supported by the research projects BFM2002-03236 of the Ministerio de Ciencia y Tecnolog¶³a, and 2002SGR 00126 of the Generalitat de Catalunya. The third author was partially supported by NSF Grant DMS-0401603. This paper was written during the third author's stay at the Centre de Recerca Matem`atica (CRM), whose support under a Mobility Fellowship of the Ministerio de Educaci¶on,Cultura y Deportes is gratefully acknowledged. It was ¯nally completed during the ¯rst and third authors' stay at the Institute for Mathematical Sciences, National University of Singapore, in July 2005. 1Throughout this paper, by \forcing" we mean \set forcing". 1 2 JOAN BAGARIA, NEUS CASTELLS, AND PAUL LARSON Given a complete Boolean algebra B in V , we can de¯ne the Boolean- valued model V B by recursion on the class of ordinals On: B V0 = ; [ B B V¸ = V¯ , if ¸ is a limit ordinal ¯<¸ B B V®+1 = ff : X ! B j X ⊆ V® g; B S B B Then, V = ®2On V® . The elements of V are called B-names. Every element x of V has a standard B-name x·, de¯ned inductively by: ;· = ;, and x· : fy· : y 2 xg ! f1Bg. B B For each x 2 V , let ½(x) = minf® 2 On j x 2 V®+1g, the rank of x in V B. Given ', a formula of the language of set theory with parameters in V B, we say that ' is true in V B if its Boolean-value is 1B, i.e., V B ² ' i® [[']]B = 1B; where [[¢]]B is de¯ned by induction on pairs (½(x); ½(y)); under the canonical well-ordering of pairs of ordinals, and the complexity of formulas (see [4]). V B can be thought of as constructed by iterating the B-valued power-set B B operation. Modulo the equivalence relation given by [[x = y]] = 1, V® is B precisely V® in the sense of the Boolean-valued model V (see [4]): Proposition 1.1. For every ordinal ®, and every complete Boolean algebra B V B B B, V® ´ (V®·) , i.e., for every x 2 V , B B B B (9y 2 V® [[x = y]] = 1) i® [[x 2 V®·]] = 1 : Corollary 1.2. For every ordinal ®, and every complete Boolean algebra B, B B V® ² ' i® V ² \V®· ² '": Notation: i) If P is a partial ordering, then we write V P for V B, where B = r:o:(P) is the regular open completion of P (see [4]). M B ii) Given M a model of set theory, we will write M® for (V®) and M® B M M B for (V® ) = (V®) . iii) Sent will denote the set of sentences in the ¯rst-order language of set theory. iv) T [f'g will always be a set of sentences in the language of set theory, usually extending ZFC. v) We will write c.t.m. for countable transitive 2-model. vi) We will write c.B.a. for complete Boolean algebra. vii) For A ⊆ R, we write L(A; R) for L(fAg [ R), the smallest transitive model of ZF that contains all the ordinals, A, and all the reals. As usual, a real number will be an element of the Baire space N = (!!;¿), where ¿ is the product topology, with the discrete topology on !. Thus, the set R of real numbers is the set of all functions from ! into !. Throughout this paper, we often talk in terms of generic ¯lters instead of Boolean-valued models. Each way of talking can be routinely reinterpreted in the other. AN ­-LOGIC PRIMER 3 Let P be a forcing notion. We say thatx _ is a simple P-name for a real number if: i) The elements ofx _ have the form ((n;· m); p) with p 2 P and n; m 2 !, so that p °P x_(·n) =m · . ii) For all n 2 !, fp 2 P j 9m such that ((n;· m); p) 2 x_g is a maximal antichain of P. For any forcing notion P and for all P-names ¿ for a real, there exists a simple P-namex _ such that °P ¿ =x _. Hence, any P-generic ¯lter will interpret these two names in the same way. ! ! Let WF := fx 2 ! j Ex is well-foundedg, where given x 2 ! , Ex := f(n; m) 2 ! £ ! j x(¡(n; m)) = 0g, with ¡ some ¯xed recursive bijection 1 between ! £ ! and !. Recall that WF is a complete ¦1 set (see [4]). let T be a theory whose models naturally contain a submodel N of Peano Arithmetic. A model M of T is an !-model if N M is standard, i.e., it is isomorphic to !. In this case, we naturally identify M with its isomorphic copy M 0 in which N M 0 is !. Stationary Tower Forcing, introduced by Woodin in the 1980's, will be used to prove some important facts about ­-logic: De¯nition 1.3. (cf.[6]) (Stationary Tower Forcing) i) A set a 6= ; is stationary if for any function F :[[a]<! ![a, there exists b 2 a such that F "[b]<! ⊆ b. ii) Given a strongly inaccessible cardinal ·, we de¯ne the Stationary Tower Forcing notion: its set of conditions is P<· = fa 2 V· : a is stationaryg; and the order is de¯ned by: a · b i® [ b ⊆ [a and fZ \ ([b) j Z 2 ag ⊆ b: Fact 1.4. Given γ < ± strongly inaccessible, a = P!1 (Vγ) 2 P<±. <! <! Proof: Given F :[Vγ] ! Vγ; let x 2 [Vγ] and let: <! A0 = x; An+1 = An [ fF (y): y 2 [An] g S <! Let b = n2! An. So, b 2 P!1 (Vγ) and F "[b] ⊆ b. ¤ Recall the large-cardinal notion of a Woodin cardinal: De¯nition 1.5. ([10]) A cardinal ± is a Woodin cardinal if for every function f : ± ! ± there exists · < ± with f"· ⊆ ·, and an elementary embedding j : V ! M with critical point · such that Vj(f)(·) ⊆ M. Theorem 1.6. (cf. [6]) Suppose that ± is a Woodin cardinal and that G ⊆ P<± is a V -generic ¯lter. Then in V [G] there is an elementary embedding j : V ! M, with M transitive, such that V [G] ² M <± ⊆ M and j(±) = ±. Moreover, for all a 2 P<±, a 2 G i® j" [ a 2 j(a). 4 JOAN BAGARIA, NEUS CASTELLS, AND PAUL LARSON 1.2. De¯nition of ²­ and invariance under forcing. De¯nition 1.7. ([17]) For T [ f'g ⊆ Sent, let T ²­ ' B B if for all c.B.a. B, and for all ordinals ®, if V® j= T then V® j= '. If T ²­ ', we say that ' is ­T -valid, or that ' is ­-valid from T . Observe that the complexity of the relation T ²­ ' is at most ¦2. Indeed, T ²­ ' i® B B 8B8®(B a c.B.a. ^ ® 2 On ! (V® ² T ! V® ² ')) The displayed formula is ¦2, since to be a c.B.a. is ¦1 and the class function B ® 7! V® is ¢2 de¯nable (i.e., both §2 and ¦2 de¯nable) in V with B as a parameter. Clearly, if T ² ' then T ²­ '. Observe, however, that the converse is not true. Indeed, we can easily ¯nd ­ZFC -valid sentences that are undecidable in ¯rst-order logic from ZFC, i.e., sentences ' such that ZFC 6² ' and ZFC 6² :'. For example, CON(ZFC): For all ® 2 On and all c.B.a. B B B, if V® ² ZFC, then since V® is a standard model of ZFC, we have B V® ² CON(ZFC). Under large cardinals, the relation ²­ is absolute under forcing extensions: Theorem 1.8. ([17]) Suppose that there exists a proper class of Woodin cardinals. If T [ f'g ⊆ Sent, then for every forcing notion P, P T ²­ ' i® V ² \T ²­ '" _ Proof: )) Let P be a poset. Suppose ¯;· Q_ 2 V P are such that V P ² \V Q ² ¯· P¤Q_ P¤Q_ T ".
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