THE POINTWISE VIEW OF DETERMINACY: ARBOREAL FORCINGS, MEASURABILITY, AND WEAK MEASURABILITY BENEDIKT LOWEÄ Abstract. We prove that for all standard arboreal forcing no- tions P there is a counterexample for the implication \If A is de- termined, then A is P-measurable". Moreover, we investigate for which forcing notions this is extendible to \weakly P-measurable". 1. Introduction The use of coding techniques is ubiquitous in the theory of Determi- nacy: we code countably many reals as one, ¯nite sequences as natural numbers, and basic open sets as ¯nite sequences. A consequence of this fact is that many proofs using Determinacy don't look at the set under investigation but at some coded or decoded version. Since most of the literature on determinacy works with the assumption that a whole pointclass (which is normally respected by the coding) is determined, this is not a problem. As soon as we drop the talk about pointclasses and move on to individual sets, we start getting into trouble: If we ask whether a given determined set has nice properties (for example, the Baire property), we tend to get unpleasant answers: In general, determined sets can be as nasty as you want them to be. Determinacy is a very local property. This paper is part of a project trying to understand the actual con- sequences of determinacy better. The di®erence between pointwise and classwise views of determinacy is of importance for higher set theory (it is, e.g., one of the di®erences being homogeneously Suslin and being determined that the ¯rst property has pointwise consequences whereas the second has only classwise consequences), and the present author has used counterexamples like the ones constructed in this paper in 2000 Mathematics Subject Classi¯cation. 03E15 03E05 03E60 06A06. The author thanks the anonymous referee for critical comments on an earlier version that led to substantial changes in the second half of the paper and Philipp Rohde (Aachen) for critical proofreading. 1 2 BENEDIKT LOWEÄ [LÄo01] to show that the usual proof of Turing determinacy will not work under the assumption of imperfect information determinacy. The Baire property in the above example is just one of many regular- ity properties derived from forcing notions. The corresponding algebras and ideals have been investigated long before the advent of forcing in [Ma35], and since then they have been the subject of a consider- able amount of research. As a short reference list, we mention [Ell74], [LaRe95Ã ], [La96Ã ], [Br95], [LÄo98] and [BrLÄo99]. Among these forcing notions, we tend to distinguish between so- called \topological forcings" with the property that the forcing condi- tions form a topology base, and the \non-topological forcings". In this paper, we shall look at the interplay between determinacy and the set algebras de¯ned by those non-topological forcings. We shall show that for any given forcing notion P (subject to certain non- triviality restraints), it is not the case that any determined set A lies in the algebra of measurability de¯ned by P. Furthermore, we look at the notion of weak P-measurability (which is classwise equivalent to P-measurability) and determine which forcing notions P allow us to construct counterexamples to pointwise implications from determinacy to weak P-measurability by a Bernstein construction. 2. Definitions and Standard Constructions Since this paper is about distinguishing properties of single sets from properties of classes of sets, let's introduce a way of addressing this di®erence: If © and ª are predicates of subsets of !!, we say that © follows pointwise from ª if for all sets A the implication \ª(A) ! ©(A)" holds. We say that © follows classwise from ª if for all boldface pointclasses ¡ we have the following implication: \if we have ª(A) for all A 2 ¡, then we have ©(A) for all A 2 ¡" (where a boldface pointclass is a pointclass closed under continuous preimages; cf. [Mo80, pp. 19 & 27]). As we mentioned in the introduction, most of the work on determi- nacy and regularity properties focuses on classwise implications. We shall look at the fact that pointwise implications do not hold, i.e., we'll construct determined sets which don't have regularity properties. Obviously, in order to construct sets which don't have regularity properties, we shall need the Axiom of Choice which will be assumed tacitly throughout this paper. THE POINTWISE VIEW OF DETERMINACY 3 This paper uses the standard notation and the standard de¯nitions of set theory of the reals, [BaJu95], Descriptive Set Theory [Mo80] and the theory of forcing [Ku80]. The reals are identi¯ed either with Baire space !! or with Cantor space 2!. In general, we look at spaces N ! where 2 · N · !. 2.1. Arboreal Forcings. We look at (symmetric) regularity proper- ties derived from arboreal forcing notions (as introduced in [LÄo98]). A forcing notion P is called arboreal on N if it is isomorphic to a family of pruned trees on N ordered by inclusion. This is equivalent to saying that the forcing conditions are closed sets in N !. For each tree P we let [P ] be the set of all branches through P . For a ¯nite sequence s 2 !<!, we let [s] := fx ; s ⊆ xg be the set of all branches through the tree of all sequences compatible with s. As usual in forcing, in most situations we shall assume that the forcing notions are nonatomic, i.e., that for each P 2 P there are incompatible Q and R with Q · P and R · P . (This corresponds to the generic reals not belonging to the ground model, cf. [Ku80, Lemma VII.2.4].) Most of the forcings have an even stronger property: We call P strongly nonatomic if that for each P 2 P there are Q and R with Q · P and R · P and [Q] \ [R] = ;. If we talk about arboreal forcing notions, we always keep in mind the standard examples from [Je86], [BaJu95], [Br95], [LÄo98], and [BrLÄo99]: Cohen forcing C, Hechler forcing D, Eventually di®erent forcing E, Sacks forcing S, Miller forcing M, Laver forcing L, Willowtree forcing W, Silver forcing V, Matet forcing T, and Mathias forcing R (the de¯- nitions can be found in the above mentioned papers and monographs; for E, check [BaJu95], for W and T, check [Br95, LÄo98]). Among these, S, V and W are arboreal on 2, the others are arboreal on !. Let A be a subset of N ! and P an arboreal forcing on N. We call A weakly P-measurable if there is a P 2 P such that either [P ] ⊆ A or [P ] ⊆ N !nA. We call A P-measurable if for each Q 2 P there is a P · Q such that either [P ] ⊆ A or [P ] ⊆ N !nA. Both P-measurability and weak P-measurability are symmetric con- cepts: If A is (weakly) P-measurable, then its complement is (weakly) P-measurable as well. It is clear that P-measurability implies weak P-measurability pointwise. Brendle and the present author show in [BrLÄo99, Lemma 2.1] that the converse holds classwise. Two remarks are in order: (1) These notions of measurability do not necessarily generate a σ- algebra. For this, we need some sort of fusion property (cf. [Je86, 4 BENEDIKT LOWEÄ De¯nition I.3.6]) which holds for the mentioned standard examples. Since this is irrelevant for the investigation at hand, we shall not discuss it here. (2) Some arboreal forcing notions P generate a topology, i.e., their conditions form a topology base. In this situation, we have a di®erent candidate for a natural algebra de¯ned by P: the σ-algebra of sets hav- ing the Baire property in the topology generated by P. More often than not, this algebra is more natural than the algebra of the P-measurable sets: Look at Cohen forcing where C-measurability translates into \for each open set there is an open subset contained in A or contained in the complement of A". Clearly, the set of rationals doesn't have this prop- erty. This means that the C-measurable sets do not form a σ-algebra, and they don't include all Fσ sets. In these cases (among our standard examples, this happens for C, D, E and R) we de¯ne the corresponding regularity property to be the Baire property associated to the topology. We will not deal with a situation like this in this paper. 2.2. Determinacy. If T is a tree on N and t 2 T , we say that ² T doesn't split at t if t has exactly one immediate successor, ² T fully splits at t if for all n 2 N, tahni 2 T . A tree σ ⊆ N <! is called a strategy for player I on N if it doesn't split at any node of even length, and fully splits at every node of odd length. A tree ¿ ⊆ N <! is called a strategy for player II on N if it doesn't split at any node of odd length, and fully splits at every node of even length. A set A is called determined if it contains the branches of a strategy σ for player I (i.e., [σ] ⊆ A, in that case, we call σ a winning strategy for player I) or its complement contains the branches of a strategy ¿ for player II (i.e., [¿] ⊆ N !nA, in that case, we call σ a winning strategy for player II). A tree T on N will be called a substrategic I-tree (substrategic II-tree) if there is a strategy σ on N for player I (for player II) such that T is a perfect subtree of σ.