Forcing Closed Unbounded Subsets of ω 2

Transcription

1 Forcing Closed Unbounded Subsets of ω 2 M.C. Stanley July 2000 Abstract. It is shown that there is no satisfactory first-order characterization of those subsets of ω 2 that have closed unbounded subsets in ω 1, ω 2, and GCH preserving outer models. These anticharacterization results generalize to subsets of successors of uncountable regular cardinals. Similar results are proved for trees of height and cardinality κ + and for partitions of [κ + ] 2, when κ is an infinite cardinal. 1. Introduction Intractable trees in L Intractable partitions from intractable trees Adding intractable sets by forcing Intractable sets in L The characterization problems The tree theorem Threads and strings Decoding The forcing conditions Extension lemmas The theorem References Introduction Which unbounded subsets of ω 2 have closed unbounded (club) subsets in ω 1 and ω 2 preserving outer models? The upshot of this paper is that this question, as well as some other combinatorial questions, has no satisfactory answer. If V is a standard transitive model of ZFC, say that W is an outer model of V if W V is also a standard transitive model of ZFC and V OR = W OR. Assume the GCH, as we shall everywhere. (Nevertheless, we indicate where this assumption is actually used.) Previously known results For subsets of ω 1, the analogous question has a well known and satisfying answer due to Baumgartner, Harrington, and Kleinberg [BHK], and independently to Jensen: If S ω 1, then the following three statements are equivalent: S is stationary. Key words and phrases. stationary set, closed unbounded set, tree, partition, forcing, class forcing, morass, coding the universe AMS Subject classification. 03E05, 03E35, 03E40, 03E45 Research supported by N.S.F. Grant DMS

2 1. INTRODUCTION S has a club subset in a cardinal and GCH preserving set generic extension. S has a club subset in an ω 1 -preserving outer model. If we restrict ourselves to extensions that add no bounded subsets of ω 2, then again a satisfying characterization is well known, c.f. [AS]. If S is a set of ordinals, then for each α < sup(s) of uncountable cofinality, fix a monotonically increasing continuous sequence ρ α i : i < cf(α) cofinal in α. Define the pattern of S at α by ptn(s, α) = i < cf(α) : ρ α i } S. Modulo the non-stationary ideal on cf(α), the definition of ptn(s, α) does not depend on the choice of the sequence ρ α i : i < cf(α). Say that S κ is fat stationary if, for each uncountable regular µ < κ, } α S : cf(α) = µ and ptn(s, α) has a club subset is stationary in κ. The following three statements are equivalent: S is fat stationary in ω 2. S has a club subset in a cardinal and GCH preserving set generic extension adding no bounded subsets of ω 2. S has a club subset in an outer model with no new bounded subsets of ω 2. Indeed, assuming the GCH, this equivalence holds with ω 2 replaced by any uncountable regular κ. This is proved by Abraham and Shelah in [AS]. They attribute the case of regular κ to Stavi, as well as, in some form, to Baumgartner, Fleissner and Kunen, and to Gregory and Harrington. Dropping the restriction that no bounded subsets of ω 2 are added, our question becomes more subtle. Intractable trees, sets, and partitions Because we want to consider some related problems, let us consider adding club subsets to subsets of successors of regular uncountable cardinals in general, rather than just to subsets of ω 2. One might suppose that the more general problem is to characterize those subsets of κ + that have a club subset in a κ and κ + preserving outer model, or, perhaps, in such an outer model in which κ remains regular. It turns out that a slightly different condition is more natural. These two alternatives are discussed later in this section, where it is observed that adopting either does not change our anticharacterization results. Suppose that S is an unbounded subset of κ +, where κ ω 1 is regular (in the model V ). Say that S is satiable if there exists a κ + -preserving outer model of V in which S has a club subset and cf(κ) > ω and α < κ + : cf V (α) = κ } remains stationary in κ +. Say that S is sated in such an outer model. Satiability of subsets of successors of singular cardinals is explored in sequels to this paper, [S3] and [S4]. By [S2], if the universe is sufficiently non-minimal and S is satiable, then S is satiable in a class generic extension, in one sense of class generic. So, at least in a 2

3 1. INTRODUCTION general sense, considering satiability in class generic extensions leads to trivialities. In this case, sufficiently non-minimal is somewhat technical. An adequate condition is the following: sup(v OR) is a definably regular cardinal in the least admissible set with V as an element. Let us say that S is intractable if S is satiable, but insatiable in any set generic extension. Before outlining our results, let us introduce two related combinatorial problems. If T is a normal tree of height and cardinality κ +, say that T is satiable if T has a cofinal branch in a κ + -preserving outer model. If F : [κ + ] 2 2 is a partition, say that F is satiable if F has a homogeneous set of cardinality κ + in a κ + -preserving outer model. Say that T or F is intractable if it is satiable, but insatiable in any set generic extension. Items (1) (6) outline the results in this paper. The first three concern the existence of intractable objects. The second three address first-order characterizations of satiability. (1) In L, if κ is infinite, then there exists an intractable tree of height and cardinality κ +. If κ ω 1 is regular, then there exists an intractable stationary subset of κ +. (2) In general, if there exists an intractable tree of height and cardinality κ +, then there exists an intractable partition of [κ + ] 2. (3) In general, if κ ω 1 is regular and there exists an intractable tree of height and cardinality κ + that has a cofinal branch in some outer model having the same bounded subsets of κ +, then there exists a cofinality and GCH preserving set generic extension in which κ + has an intractable subset. The extra hypothesis in (3) that the intractable tree has a branch in an outer model with the same bounded subsets of κ + is automatically satisfied if V = L, or more generally in the presence of sufficient strong covering. Anticharacterization results The question posed at the outset is a characterization problem. The existence of intractable subsets of ω 2 does not directly address it, though it does suggest that no fixed depth of analysis in general suffices to decide whether a given stationary subset of ω 2 has a club subset in some ω 1 and ω 2 preserving outer model. In fact, the ideas used to prove items (1) (3) can be extended to show that the characterization problems have no satisfactory solution. To be more precise, let us say that the stationary set characterization problem at κ + (respectively, the partition and tree characterization problems) is to give a firstorder definition of the collection of satiable subsets of κ + (respectively, of satiable partitions and trees at κ + ). Say that the stationary set characterization problem at κ + is solvable in V if } S V : S κ + and S is satiable (with respect to outer models of V ) V. Solvability of the other characterization problems in V is defined analogously. 3

4 1. INTRODUCTION The minimum model of ZFC is the smallest transitive standard model of ZFC (provided any exist, of course). In other words, the minimum model is L α, where α is least such that L α ZFC. (4) In the minimum model of ZFC, the tree and partition characterization problems are unsolvable at all κ + and the stationary set characterization problem is unsolvable at κ + when κ ω 1 is regular. (5) All of the characterization problems are solvable in any sufficiently non-minimal model of ZFC. Again, we may take sufficiently non-minimal to mean that sup(v OR) is a definably regular cardinal in the least admissible set with V as an element. Item (5) raises the hope that the question with which we began might have a satisfactory answer, at least in sufficiently non-minimal models. Unfortunately, the solutions in these models are not parameter-free. There exist sets in such models that accidentally comprise exactly the satiable objects of various sorts. It is really uniform solutions that we want. Let us say that a characterization problem at κ + is uniformly solvable if there exists a formula, perhaps with parameters, that defines the set of satiable objects at κ + in any outer model in which κ + maintains its status as the successor of the cardinal κ (and κ remains regular, in the case of the stationary set problem). This notion is, if anything, too generous. If κ + is a definable cardinal, like ω 2, then it would be natural to ask for a parameter free definition. Modifying the definitions to allow these cases, the characterization problem for stationary subsets of ω 1 and the partition and tree problems at ω are uniformly solvable. For example, S is stationary in ω 1 can be formalized with a parameter free formula. There is no such characterization of the subsets of ω 2 that have a club subset in an ω 1 and ω 2 preserving outer model: (6) Suppose that κ is an infinite cardinal in V. There exists an outer model W of V in which all cardinals and cofinalities less than or equal to κ + are preserved and in which the tree and partition characterization problems are unsolvable. If κ ω 1 is regular in V, then the stationary set characterization problem is unsolvable in W, as well. Thus, the tree and partition characterization problems at κ + are not uniformly solvable. If κ ω 1 is regular, then the stationary set characterization problem at κ + is not uniformly solvable. A caveat regarding (6) is that the outer model W is of the form L[A], where A κ +. Thus (6) precludes uniform solutions for outer models satisfying ZFC, but not necessarily for outer models satisfying extensions of ZFC such as ZFC + there exists a strongly compact cardinal. This observation points out that, negative though these results may seem, they do leave tantalizing possibilities open. Returning to the question with which we began, suppose that robust and stunted are two first-order parameter free properties such that if S ω 2 is robust, then S has a club subset in an ω 1 and ω 2 preserving outer model of ZFC + GCH and 4

5 1. INTRODUCTION if S ω 2 is stunted, then S does not have a club subset in any ω 1 and ω 2 preserving outer model of ZFC + GCH. Our results imply that no matter what our choice of the properties robust and stunted, there exist models in which there are S ω 2 that are neither robust nor stunted. Question 1.1. Do there exist such properties robust and stunted and a model in which every S ω 2 is either robust or stunted? If so, this provides a uniform characterization of satiable subsets of ω 2 relative to a consistent extension of ZFC + GCH, namely, ZFC + GCH + every subset of ω 2 is either robust or stunted. That is cheating, though. In this vein an even more egregious fraud is ZFC + V =L, which precludes all proper outer models. The question, then, is whether there is a uniform characterization relative to a natural extension of ZFC + GCH that does not trivialize outer models. For example, one might ask Question 1.2. Does there exist a large cardinal property ϕ such that, working exclusively with models (and outer models) of ZFC + GCH + κ ϕ, the set of satiable subsets of ω 2 is uniformly characterizable? Sy Friedman in [F2] has used some of the results in this paper together with machinery from his work on generic Π 1 2-singletons to obtain several interesting results on constructibility degrees. His are first-order theorems of ZFC + 0 # exists. Among his results are those in the following Theorem 1.3. (Sy Friedman) Assume that 0 # exists and that κ is an L-definable L-successor cardinal. Then the following sets are equiconstructible with 0 # : (1) T L : T is a tree of height and cardinality κ having a cardinal-preserving branch } (2) S L : S κ having a cardinal-preserving club subset }, where κ > ω L 1 is the L-successor of an L-regular cardinal. The L-definability of κ can be dropped if branches and club subsets are allowed to belong to a set-generic extension of V. Alternative definitions of satiable set It is perhaps unclear that we have chosen a good definition of satiable for subsets of κ + when κ > ω 1 is regular. The purpose of this tangential subsection is to consider some alternatives. Though some questions are open regarding implications between various of these properties, the anticharacterization results of this paper cannot be ameliorated by choosing any of them. The final remark in this subsection is that the characterization problem for subsets of κ + of bounded pattern width is easy when κ is regular. This problem is difficult but solvable at ℵ ω+1 and is unsolvable at ℵ ω1 +1. Let us make some temporary definitions that will be abandoned at the end of this subsection. Assume that κ is a regular uncountable cardinal in the model V. Recall that S κ + is satiable if it has a club subset in a κ + -preserving outer model in which cf(κ) > ω and α < κ + : cf V (α) = κ } remains stationary. Say that S κ + is satiable 1 if there exists an outer model of V in which S has a club subset and both κ and κ + remain cardinals. Say that S κ + is satiable 2 if there exists an outer model of V in which S has a club subset and both κ and κ + remain regular. 5

6 1. INTRODUCTION Remark 1.4. If κ = ω 1, then satiable, satiable 1, and satiable 2 are equivalent. Remark 1.5. If we require that covering holds between the inner model V and the outer model in which S has a club subset, then satiable 1 and satiable 2 are equivalent and either implies satiable. Remark 1.6. In general, satiable 2 implies satiable 1, but not conversely. Proof: An example of a set that is satiable 1 but not satiable 2 is S = } α < κ + : cf(α) < κ when κ is a measurable cardinal. After adding a Prikry sequence, S is fat stationary, hence has a club subset in a cardinal preserving outer model. I do not know whether satiable implies satiable 1, even assuming covering. Answering the following question is perhaps the place to begin: Question 1.7. Assume V = L. Suppose that S ω 3 has a club subset in an ω 1 and ω 3 preserving outer model in which ω2 L has cofinality ω 1 and the set of ordinals of L-cofinality ω2 L remains stationary in ω3 L. Does S have a club subset in some (other) ω 2 and ω 3 preserving outer model? Remark 1.8. The anticharacterization results (4) (6) of this paper still hold if we replace satiable with either satiable 1 or satiable 2. Proof: Result (4) is that stationary set characterization problem is unsolvable in the minimum model. In this setting, satiable 1 and satiable 2 both imply satiable. The proof of (4), which appears in 5 and Example 1 of 6, proceeds as follows: Given a parameter free sentence ϕ of the language of set theory, a subset X ϕ κ + is constructed. If L ϕ, then X ϕ has a club subset in a cofinality and GCH preserving class generic extension of L, hence X ϕ is satiable 1 and satiable 2, as well as satiable. Conversely, if X ϕ is satiable (hence, if X ϕ is satiable 1 or satiable 2 ), then L ϕ. Because ϕ : L ϕ } / L when L is the minimum model, this suffices. Result (5) is that the stationary set characterization problem is solvable in any sufficiently non-minimal model. This is a soft result in infinitary logic, found in Example 2 of 6. It has the same proof with satiable 1 or satiable 2 in place of satiable. Finally, result (6) is that there is no uniform solution to the stationary set characterization problem. This appears as Example 3 in 6. When reading it, note that in the cycle of equivalences (1) (7) that show the tree T ϕ is satiable iff the set S ϕ is satiable, line (6) implies that S ϕ is satiable 2, hence also satiable 1. The purpose of the following remark is seeing that satiable extracts the uncharacterizable part of the stationary set problem. For this we need some more terminology. Again, this extra terminology will be abandoned at the end of this subsection. If ω 1 µ κ and µ is regular, say that an unbounded S κ + is µ- satiable if S has a club subset in a κ + -preserving outer model in which cf(µ) > ω and α < κ + : cf V (α) = µ } is stationary in κ +. Satiable is κ-satiable in this terminology. 6

7 1. INTRODUCTION Remark 1.9. Suppose that S is an unbounded subset of κ +. The following two statements are equivalent. (1) S has a club subset in a κ + -preserving outer model in which κ + ℵ 2. (2) S is µ-satiable, for some regular µ κ. Proof: (2) (1) is immediate from the definition. For the converse, suppose that S has a club subset in a κ + -preserving outer model in which κ + ℵ 2. Ordinals of uncountable cofinality in that outer model are stationary in κ +, and these ordinals are partitioned into at most κ many V -cofinalities. Remark (GCH) Suppose that µ is uncountable and regular, that µ < κ, and that µ cf(κ). For S κ +, S is µ-satiable is equivalent to a first-order property. Proof: Under these hypotheses the following three statements are equivalent: (1) S is µ-satiable. (2) α S : cf(α) = µ and ptn(s, α) is stationary in µ } is stationary in κ +. (3) S has a club subset in a set forcing extension in which µ is Lévy collapsed to ω 1 and κ + to ω 2. For (2) (3), note that since 2 µ < κ +, there exists a fixed stationary pattern at cofinality µ points in S that occurs stationarily in κ +. After Lévy collapsing µ to ω 1 and κ + to ω 2, force to add a club subset to this pattern. Then S is fat stationary. Thus it is only the κ-satiability characterization problem that does not admit an easy solution. At this point a remark motivated by results in [S2] and [S3] is an easy observation. These papers consider satiability of subsets of successors of singular cardinals. In this setting sets of bounded pattern width are important. If S is a subset of κ + > ω 1, say that S has bounded pattern width if fewer than the largest possible number of patterns occur in S, that is, if sup ω 1 τ<κ + τ regular P τ < sup ω 1 τ<κ + τ regular where P τ consists of those equivalence classes, modulo the non-stationary ideal, that have representatives in 2 τ, } ptn(s, α) : α lim(s) and cf(α) = τ. We assume the GCH everywhere, so in the context of this paper S κ + has bounded pattern width when the quantity on the right hand side of this inequality is κ +. In the case of regular uncountable κ, it is easy to give a first-order characterization of the bounded pattern width subsets of κ + that are satiable: 7

8 1. INTRODUCTION Remark (GCH) Suppose that κ is regular and S is a bounded pattern width subset of κ +. Then the following statements are equivalent: (1) S is satiable. (2) There exists a regular uncountable µ κ and a stationary set X µ such that } α S : cf(α) = µ and ptn(s, α) = X (mod NS) is stationary in κ +. (3) S has a club subset in a GCH preserving set forcing extension in which κ + = ω 2. Proof: This is essentially the same argument as for Remark For (1) (2), using that S has bounded pattern width, note that in an outer model in which S is sated some fixed pattern occurs (mod NS) stationarily often in S at ordinals of outer model cofinality ω 1. For (2) (3), first Lévy collapse µ to ω 1 and κ + to ω 2. Add a club subset to X by forcing. Then S is fat stationary. Outline Each of 2 7 can be read independently. Section 2. It is not difficult to define in L an intractable tree of height and cardinality κ + and to show that it is insatiable in any set generic extension. This is done in 2. Showing that this tree is satiable in a class generic extension uses an adaptation of ideas used in coding the universe and is more involved. This work is relegated to 7. Section 3. Using standard ideas, this short section proves that intractable partitions are definable from intractable trees. It follows that intractable partitions exist in L. Section 4. This section contains two results. The first is the Stationary Set Theorem: Given a partition F : [κ + ] 2 2, a (set) forcing property is defined that adds a stationary S κ + such that if α < β < κ + both have cofinality κ, then ptn(s, α) and ptn(s, β) are compatible iff F (α, β) = 1. Incompatible patterns cannot both have club subsets in a κ-preserving outer model. The forcing is designed so that if H is F -homogeneous to 1, then ptn(s, α) : α H } generates a normal filter on κ. The Intractable Set Theorem uses this to show that a model containing an intractable tree of height and cardinality κ + can be extended (by set forcing) to obtain a model with an intractable subset of κ +, assuming the intractable tree has a cofinal branch in some strongly κ + -covered outer model. Section 5. A universal (κ, 1)-morass is used to construct an intractable subset of κ + from an (appropriate) intractable partition on κ +. It follows that intractable sets exist in L. The morass is used directly in a somewhat unusual way. I do not know how to use a morass to construct a suitably generic object for the forcing of 4. Section 6. This section proves three results to justify claims (4) (6) enumerated earlier. The first of these uses a minor modification of the intractable tree defined in 2. It shows that the tree characterization problem is unsolvable in the minimum model. The second uses infinitary logic to show that the characterization problems are solvable in any sufficiently non-minimal model. 8

9 1. INTRODUCTION The third uses an application from [BJW] ( minimalizing the universe ) to show that the characterization problems are not uniformly solvable. Section 7. This long section, broken into five subsections, develops a version of coding the universe to finish the proof in 2 that intractable trees exist in L. Rather than outlining modifications to Jensen s proof in [BJW], a proof starting from scratch and using Easton support coding is given. 2. Intractable trees in L The following theorem underlies all of the anticharacterization results in this paper. Tree Theorem. (V =L) Suppose that κ is an infinite cardinal. There exists a normal tree T of height and cardinality κ + with the following four properties. (1) T has no cofinal branch in any set generic extension in which κ + remains a cardinal. (2) There exists a cardinal-preserving L-definable class forcing P that adds a cofinal branch to T and adds no bounded subsets of κ +. (3) There exists a bijection g: T κ + such that if µ κ is infinite and regular and if b is a P generic branch through T, then in L[b] g(t) : t b and cf ( g(t) ) } = µ is stationary in κ +. (4) Suppose that κ < α κ +, that α is admissible or is a limit of admissibles, and that L α x x κ. Then T L α and g (T L α ) are (uniformly) Σ 1 definable over L α. In principle this theorem is known to those familiar with class generic Π 1 2 singletons: Apply David s trick to Jensen coding of the universe with a subset of κ +. Rather than outlining modifications to Jensen s theorem, however, we shall give a detailed proof using simplified coding techniques. This coding method, though adequate for our purpose here, destroys most large cardinal properties. After coding no Mahlo cardinals remain. In this section we define the tree T and verify (1) and the definability claim regarding T in (4). The heavy lifting required for claims (2) and (3), as well as the definability claim regarding g in (4), is relegated to 7. Say that an ordinal has color δ if it lies in the class Z δ = } <δ, > : OR, where <, > indicates Gödel pairing. Work in L. If ν is an infinite cardinal, let C ν consist of sets c [ν, ν + ) such that c is bounded below ν + and contains only ordinals of color <3, 0>. Ordinals of certain other colors are reserved for use in coding. Let C ν be ordered by reverse end extension. Then C ν is just ν + -Cohen forcing restricted to ordinals in [ν, ν + ) of a particular color 9

10 2. INTRACTABLE TREES IN L (and is equivalent to ν + -Cohen forcing). A bit abusively, say that a set or class of ordinals X is C ν generic if c C ν : c = X [ν, α) Z <3,0>, for some α < ν + } is C ν generic, i.e., is a filter meeting every dense subset of C ν that lies in L. Given b: [κ, κ + ) 2, in 7.2 we shall specify a uniformly 2 definable class of ordinals Decode(b) obtained by recursively (with respect to L-cardinals) decoding b. Abstractly, the plan for decoding b is the same as with Jensen coding. From b a subset of [κ +, κ ++ ) L is recovered using a variation on ordinary almost disjoint coding. In turn a subset of [κ ++, κ +++ ) L is recovered from this set, and so forth. Rather than almost disjoint families of sets of ordinals, suitably almost disjoint families of threads are used, so that the same procedure can be used at limit cardinals. In order to keep the coding at one cardinal from interfering with that at others, different colors of ordinals are used for coding different intervals. At inaccessibles, the decoding must simultaneously recover the assignment of colors. Assuming this notion, we can define the tree T described in the Tree Theorem. Let ZF 2 be ZF with collection restricted to Σ 2 formulas and separation restricted to Σ 1 formulas. If L α satisfies ZF 2, then L α satisfies choice, as well. Furthermore, L α satisfies ZF 2 iff L α is power admissible iff L α -cardinals are unbounded in α and (L α ; CARD L α ) is admissible. It is because decoding is recursive relative to a predicate for L-cardinals that it is convenient to work with ZF 2 -ordinals. Declare that t T iff t: [κ, α) 2, for some α < κ +, and if, whenever α and L β ZF 2 + = κ +, we have that either L β [t ] is not admissible or L β [t ] Decode ( t ) is C ν generic over L β, for all L β -cardinals ν κ. T is ordered by functional extension. Lemma. T is a normal tree of height and cardinality κ +. Proof: Certainly T has cardinality at most κ +. To see that T has height κ +, let α < κ +. We may assume that α κ + κ, that L α x x κ, and that there exists an L α -definable function mapping κ cofinally into α. (For example, given an ordinal ᾱ between κ and κ +, let α > ᾱ be least such that L α Σ1 L κ +.) Let f: κ α be a bijection, and let t: [κ, α) 2 be such that t(ζ) = 1 iff ζ = κ + <δ, γ>, for some δ, γ < κ such that f(δ) < f(γ). Suppose that α and that L β ZF 2 + = κ +. Then > κ + κ and β α by our choice of α. There is a well ordering of type α in L β [t ], hence this structure is not admissible. Thus t T. Similarly, T is a normal tree. Lemma. Suppose that κ < α κ +, that α is admissible or is the limit of admissibles, and that L α x x κ. Then T L α is (uniformly) Σ 1 definable over L α. Proof: Note that T L α can be defined over L α by a formula ( δ t: [κ, δ) 2 γ ( L γ δ κ δ β γ ϕ(t,, β) )), where ϕ is Σ 1. 10

11 2. INTRACTABLE TREES IN L Lemma. Suppose that (κ + ) L is a cardinal and that b: [κ, κ + ) 2 is a cofinal branch through T. Then b is not set generic over L. Proof: Because no set forcing over L adds C L ν generic objects for unboundedly many ν, it suffices to see that such generic objects lie in L[b]. If ν κ is an L- cardinal such that Decode(b) is not C L ν generic over L, then working in L[b] we can choose M Σ2 L[b] such that M < (κ + ) L and κ κ, b, ν, D} M, where D is a constructible dense subset of C L ν such that M Decode(b) fails to meet D. Say = M κ + and π: L β [b ] = M. Then L β [b ] satisfies ZF 2 + = κ + + Decode(b ) fails to meet π 1 (D), contradicting that b T. 3. Intractable partitions from intractable trees The following theorem shows that if there exists an intractable tree on κ +, then there exists an intractable partition. The ideas used in its proof are standard. Theorem. Suppose that T is a normal rooted tree of height and cardinality κ +. Then there exists a partition F : [κ + ] 2 2 such that in any κ + -preserving outer model, the tree T has a cofinal branch iff the partition F has a homogeneous set of cardinality κ +. Proof: Let < T be the tree ordering on T and let ht(t) indicate the height of the node t in this ordering. Let < be a linear ordering of T with the property that there are no monotonically < -increasing or < -decreasing sequences of length κ + in any κ + -preserving outer model. The lexicographic order on κ κ has this property. Define a linear ordering extending < T by declaring that t t iff t < T t or t < t, where t is < T -least such that t T t but t T t, and t is < T -least such that t T t but t T t. Let f: κ + T be a bijection and declare that F ( α, β} ) 0, if α < β and f(α) f(β) = 1, if α < β and f(β) f(α) Set µ = κ + and work in a µ-preserving outer model. (For clarity, we introduce µ since κ may not be preserved.) Then µ is a successor cardinal, hence regular and uncountable. If T has a cofinal branch, then F has a set homogeneous to 0 of cardinality µ. On the other hand, suppose that H µ is F -homogeneous and has cardinality µ. Let t i : i < µ be such that t i f H and both ht(t i ) ht(t j ) and f 1 (t i ) < f 1 (t j ) when i < j. Fix i < µ. For j > i, let t j be the greatest t such that t T t i and t T t j. Note that t j T t j, for all j and j such that i < j < j. Indeed, if t j < T t j, then t j t j iff t j t i, contradicting that H is homogeneous. It follows that t j = t j, for all sufficiently large j and j. We may assume, then, that for each i < µ there exists t i T t i such that t i is the greatest common T -predecessor of both t i and t j, for all j > i. Then t i and t j are T -comparable, for all i and j. It follows that if ht( t i ) = ht( t j ), then t i = t j. 11

12 3. INTRACTABLE PARTITIONS FROM INTRACTABLE TREES Finally, note that if δ < µ, then ht( t i ) > δ, for all sufficiently large i. Indeed, were t i = t for µ many i, then the immediate < T -successors of t would include a < -increasing or < -decreasing µ-sequence, depending as H is homogeneous to 0 or to 1. Thus t i : i < µ determines a cofinal branch through T. 4. Adding intractable sets by forcing Fix a regular cardinal κ ω 1 and suppose that κ holds. Let C α : α lim(κ + ) be a κ -sequence and let ρ α i : i < κ enumerate the limit points of C α in increasing order when cf(α) = κ. Note that if α, β κ + have cofinality κ and ρ α i = ρβ j, then i = j. Note also that if d is a set of fewer than κ many cofinality κ ordinals below κ +, then there exists i < κ such that if α and β are distinct elements of d, then C α \ ρ α i and C β \ ρ β i are disjoint. Indeed, given α < β, let i(α, β) be the least i such that ρ β i α. Then set i = sup α,β} [d] 2 i(α, β). If S κ +, let us use these sequences ρ α i : i < κ to define the pattern of S at α: } ptn(s, α) = i < κ : ρ α i S. Stationary Set Theorem. (2 <κ = κ) Assume that κ ω 1 is regular and that κ holds. Suppose that F : [ α < κ + : cf(α) = κ } ] 2 2. There exists a partial ordering S such that (1) S has cardinality κ +, is <κ-closed, and satisfies the <κ + -c.c. (2) Forcing with S adds a stationary set S G κ + such that (a) if α < κ + and cf(α) = κ, then α S G ; (b) if F (α, β) = 0, then ptn(s G, α) ptn(s G, β) < κ; and (c) if H is F -homogeneous to 1, then ptn(s G, α) : α H } generates a normal filter of stationary subsets of κ. Proof: Let S consist of all triples s = (d s, β s, f s ) such that d s α < κ + : cf(α) = κ } and d s < κ; β s < κ and ( ) ( ) C α \ ρ α β s Cα \ ρ α β s =, for all α, α } [d s ] 2 ; and f s : α d s ρ α i : i < β s } 0, 1}. Say that s s when d s d s and β s β s and f s f s ; and ( ) if α, α } [d s ] 2 and F (α, α ) = 0, and if f s ρ α i = 1, for some i such that ( ) β s i < β s, then f s ρ α i = 0. Note that S = κ +. Note also that S is <κ-closed. Indeed, if s i : i < θ is a descending sequence of conditions, where θ < κ, then ( inf s i = d si, sup β si, ) f si i<θ i<θ i<θ i<θ 12

13 extends every s i. If G is S generic, set S G = 4. ADDING INTRACTABLE SETS BY FORCING ρ < κ + : cf(ρ) = κ or f s (ρ) = 1, for some s G Extension Lemma. Suppose that s S. (a) If δ < κ, then s has an extension s such that β s > δ. (b) If α < κ + has cofinality κ, then s has an extension s such that α d s. Proof: We prove parts (a) and (b) simultaneously. Suppose that δ < κ and that α < κ + has cofinality κ. We may assume that α / d s. Set d = d s α}. Choose β such that β s, δ < β < κ and ( ( C α \ ρβ) α Cγ \ ρ γ β) =, for every γ d s. By setting f(ρ) = 0 when ρ is not in the domain of f s, extend f s to f: ρ γ i : i < β and γ d } 0, 1}. Then (d, β, f) is a condition. To see that (d, β, f) s, note that if γ d s and β s i < β, then ρ γ i / dom(f s ). This is because if ρ γ i = ργ j, then j = i β s. Hence f(ρ γ i ) = 0. Antichain Lemma. Every subset of S of size κ + has a subset of size κ + in which every two conditions are compatible. Proof: Suppose that A S has cardinality κ +. We may assume that (1) if s, s A, then β s = β s ; (2) there exists r κ + such that dom(f s ) dom(f s ) = r, for all s, s A such that s s ; and (3) if s, s A and ρ r, then f s (ρ) = f s (ρ). Now suppose that s, s A. Set d = d s d s and choose β β s = β s such that C α \ρ α β and C α \ ρ α β, for all α α that lie in d. Let f: ρ α i : α d and i < β } 0, 1} by setting f ( ρ α i ) = f s ( ρ α i ), if α ds and i < β s f s ( ρ α i ), if α ds and i < β s 0, otherwise. Using (2) and (3), the first two lines in the definition of f do not conflict. It follows that (d, β, f) S. Towards seeing that (d, β, f) s, suppose that α d s and that β s i < β. Note that ρ α i / dom(f s ) dom(f s ) because if ρ α i = ρ α j, then j = i β s = β s. Hence f ( ) ρ α i = 0. It follows that (d, β, f) s. Similarly (d, β, f) s. Filter Lemma. Suppose that G is S generic. (a) If F (α, α ) = 0, then ptn(s G, α) ptn(s G, α ) < κ. (b) If H lies in the generic extension by G and is F -homogeneous to 1, then ptn(s G, α) : α H } is contained in a normal filter of stationary subsets of κ. Proof: For (a), let s G be such that α, ( α ) d s. If s extends ( ) s and β s i < β s, then it is not the case that both f s ρ α i = 1 and fs ρ α i = 1. Hence s ptn( S G, ˇα) ptn( S G, ˇα ) ˇβ s. 13 }.

14 4. ADDING INTRACTABLE SETS BY FORCING For (b), it suffices to see that if α i : i < κ enumerates a subset of H (in the generic extension), then i<κ ptn(s G, α i ) is stationary in κ. Work in the ground model. Suppose that S H is ˇF -homogeneous to 1 and that S α i H, for i < κ. Fix a condition s and a term C for a club subset of κ. Define a descending sequence s n : n < ω of condition by recursion. Begin by setting s 0 = s. Then choose s n+1 to be a condition s s n such that s C [ β sn, β s ) and there exists γ i : i < β sn such that, for each i < β sn, s α i = ˇγ i and γ i d s. Note that there exists such a condition s, using the extension lemma and that S is <κ-closed. Set s = inf n<ω s n. Then s ˇβ s C and there exists γ i : i < β s such that γ i d s and s α i = ˇγ i, for all i < β s. Next define (d, β, f). Set β = β s + 1 and d = d s. Let by f ( ρ α i f: ρ α i : α d and i β s } 0, 1} ) = f s ( ρ α i ), if i < βs 1, if i = β s and α γ j : j < β s } 0, if i = β s and α d s \ γ j : j < β s }. Note that f is well defined because ρ α β s ρ α i, for i < β s, and because C α \ ρ α β s and C α \ ρ α β s are disjoint, if α α lie in d s. It follows that (d, β, f) is a condition in S. Now (d, β, f) ˇβ s ptn( S G, ˇγ i ), for each i < β s, so it suffices to see that (d, β, f) s. Suppose that α, α } [ ] 2, d s that F (α, α ) = 0, and that f ( ) ρ α β s = 1. Then α = γ i, for some i < β s. Because s forces that γ i : i < β s } H G and that H G is F -homogeneous to 1, it follows that α d s \ γ i : i < β s }. Hence f ( ρ α β s ) = 0, as required. This completes the proof of the Stationary Set Theorem. Finally, we turn to the Intractable Set Theorem. Suppose that κ ω 1 is regular in V. If there exists an intractable tree of height and cardinality κ + that has a cofinal branch in an outer model having the same bounded subsets of κ +, then there is a cofinality and GCH preserving set generic extension of V in which there exists an intractable subset of κ +. The extra hypothesis that the intractable tree has a branch in an outer model with the same bounded subsets of κ + is automatically satisfied in many circumstances. Say that W is a strongly µ-covered outer model of V, if, given any λ µ and any f and X in W such that f: λ <ω λ and X λ and X W < µ, there exists Y V such that Y X and Y W < µ and f Y <ω Y. For example, if V = L and µ ω 2 is a successor cardinal in L, then any µ-preserving outer model is strongly µ-covered. This is because such an outer model satisfies 0 # does not exist L-successor cardinals are collapsed in the presence of 0 #. Hence Shelah s strong covering holds between L and such an outer model. 14

15 4. ADDING INTRACTABLE SETS BY FORCING Remark. Assume the GCH in V. If (in V ) T is a tree of height and cardinality µ that has a new branch in a µ-preserving strongly µ-covered outer model, then T has a new branch in an outer model of V with the same bounded subsets of µ. Proof: Suppose that W is a µ-preserving strongly µ-covered outer model of V in which T has a new branch b. Because V satisfies the GCH, we may assume that V = L[A], where A is a V -amenable class of ordinals. We maintain that W = L[A, b] has the same bounded subsets of µ. Suppose that a α < µ and a L δ [A, b]. By strong µ-covering, there exists M L δ [A, b] such that α a} M and M OR V and M W < µ. Say π: M = L β [A, b γ], where γ β < µ. Now b γ V and π OR V. Hence A = π (A M) V. It follows that a V. Proof of the Intractable Set Theorem: Let b be a branch through the intractable tree T in an outer model W having the same bounded subsets of κ +. Then W is a κ and κ + -preserving outer model of V, so κ + is unambiguous. Let F be the partial ordering consisting of all one-to-one functions from ordinals less than κ + into T (ordered by reverse functional extension). Then F V = F W. Let f: κ + T be a bijection that is F generic over W. Then in W [f] the set α < κ + : f(α) b and cf(α) = κ } is stationary in κ +. Similarly, we may assume that κ holds in V by adding a κ -sequence that is set generic over W (hence over V ), if necessary. Let S be the partial ordering of the Stationary Set Theorem, where the partition F of cofinality κ ordinals below κ + is given by F (α, β) = 1 iff f(α) < T f(β) or f(β) < T f(α). Note that S V [f] = S W [f], again because conditions are small sets. Let S κ + be S generic over W [f] (a model of κ + 2 <κ =κ). A fortiori, S is S generic over V [f]. The model V [f, S] is a cofinality and GCH preserving set generic extension of V. We maintain that S is intractable (with respect to outer models of V [f, S]). Note first that if S has a club subset C in a κ + -preserving outer model of V [f, S] in which cf(κ) > ω and α < κ + : cf V [f,s] (α) = κ } is stationary in κ +, then } α lim(c) : cf V (α) = κ is F -homogeneous to 1 and unbounded in κ +. Hence T has a cofinal branch. It follows that S is not sated in any set generic extension of V [f, S], since T is an intractable tree. On the other hand, S does have a club subset in a cofinality preserving extension of W [f, S]. To see this, note first that ptn(s, α) : cf(α) = κ and f(α) b } is contained in a normal filter F of stationary subsets of κ. Thus, using [AS], we can add a club K κ that is eventually contained in each element of F, while preserving cardinals and that 2 κ = κ +. [For the reader s convenience, let us define this forcing. Declare that (r, X) R iff r is a bounded closed subset of κ and X F. Say that (r, X) (r, X ) iff r end-extends r, X X, and r \ r X. Then R κ +, since 2 κ = κ +. Using that F is a normal filter, this forcing is <κ-distributive, and, using that 2 <κ = κ, it satisfies the <κ + -c.c.] After forcing with R over W [f, S], the set S is fat stationary, and so further cofinality preserving forcing adds a club subset to S. 15

16 5. Intractable sets in L 4. ADDING INTRACTABLE SETS BY FORCING In this section we prove that in L there exists an intractable subset of κ + whenever κ ω 1 is regular. Rather than working directly with the hypothesis V =L, we shall use a universal (κ, 1)-morass. Our notation follows [D]. Theorem 5.1. Suppose that κ ω 1 is regular, that F : [ < κ + : cf() = κ } ] 2 2, and that there exists a universal (κ, 1)-morass and, for α S 0, partitions F α : [S α ] 2 2 such that F κ [ < κ + : cf() = κ } ] 2 = F and if τ < τ < ν and ν i ν and π νν ( τ) = τ and π νν ( τ ) = τ, then F α ν ( τ, τ ) = F αν (τ, τ ). Then there exists a stationary X κ + such that if < κ + and cf() = κ, then X; if F (τ, ν) = 0, then ptn(x, τ) ptn(x, ν) is non-stationary; and if H is F -homogeneous to 1, then there exists a normal filter F of stationary subsets of κ such that ptn(x, α) F, for all α H. Before proving the theorem, note this principal Corollary 5.2. (V =L) If κ ω 1 is regular, then there exists an intractable subset of κ +. Proof: The proof uses a little more than that in L there exists an intractable tree at κ + and a universal (κ, 1)-morass. It also uses that the comparability partition for that tree coheres with the natural morass as in the theorem. Let the tree T of height and cardinality κ + and the bijection g: T κ + be as in the Tree Theorem. Define 1 if g F (α < β) = 1 (α) < T g 1 (β) 0 otherwise. More generally, if α S 0 and ν = sup(s α ) and τ, τ S α ν, set 1 if F α (τ < τ Lν g 1 (τ) < T g 1 (τ ) ) = 0 otherwise, where it is the morass of [D] that provides S 0, S α, and so forth. Using that T and g are uniformly Σ 1 definable over admissibles and limits of admissibles, and that morass embeddings are Σ 1 elementary, the partitions F α cohere with the morass as required for the theorem. Using these partitions, let X be provided by Theorem 5.1. If X has a club subset in a κ + -preserving outer model in which cf(κ) > ω and α < κ + : cf L (α) = κ } is stationary in κ +, then there exists an unbounded subset of κ + that is F -homogeneous to 1 in that model. Hence T has a cofinal branch. It follows that X is not sated in any set generic extension of L. 16

17 5. INTRACTABLE SETS IN L On the other hand, T has a cofinal branch b in a P generic extension. extension, } α < κ + : cf(α) = κ and g 1 (α) b In this is stationary. Furthermore, forcing with P adds no bounded subsets of κ +, so } ptn(x, α) : cf(α) = κ and g 1 (α) t generates a normal filter of stationary subsets of κ, for all initial segments t of b. It follows that in a P generic extension, there exists a normal filter F of stationary subsets of κ such that ptn(x, α) F, for all α such that g 1 (α) b. After cofinality and GCH preserving forcing to add a club subset of κ that is eventually contained in every member of F, the set X becomes fat stationary, hence has a club subset in a further cofinality and GCH preserving forcing extension. Proof of Theorem 5.1: If ν lim(s α ), set C ν = ρ S α ν : ρ = sup ( π νν ν ) }, for some ν i ν Then C ν lim(s αν ) ν is closed; if ν S κ has cofinality κ, then C ν is unbounded in ν; ot(c ν ) α ν ; (Level Coherence) if τ C ν, then C ν τ = C τ ; and (Tree Coherence) if ν i ν and ρ = sup ( π νν ν ), then C ν ρ = sup ( π νν ρ ) : ρ C ν }. Proof of Tree Coherence: Suppose first that λ C ν ρ. Say λ = sup ( π τν τ ), where τ i ν. Then τ i ν, since λ < ρ = sup ( π νν ν ). Set ρ = sup ( π τ ν τ ). Then λ = sup ( π τν τ ) = sup ( π νν (π τ ν τ) ) = sup ( π νν ρ ). It follows that ρ < ν, or else λ = sup ( π νν ν ) = ρ. But λ C ν ρ. Thus ρ C ν and λ = sup ( π νν ρ ). Conversely, suppose that ρ C ν. Say ρ = sup ( π τ ν τ ), where τ i ν. Then sup ( π νν ρ ) = sup ( π νν (π τ ν τ) ) = sup ( π τν τ ). Because ρ < ν and π νν is order preserving, sup ( π νν ρ ) < sup ( π νν ν ) = ρ ν. It follows that sup ( π νν ρ ) C ν ρ. 17

18 5. INTRACTABLE SETS IN L Let ρ ν i : i ot(c ν ) enumerate C ν ν} monotonically. Set E α = } ν lim(s α ) : ot(c ν ) = α. Then points of cofinality κ in S κ lie in E κ. Note also that if ν E α, then ν / C τ, for all τ S α. Otherwise, ot(c τ ) > α ν. We now define sets X α S α by recursion on α. The set X that we seek is First set Y α = X = X κ < κ + : cf() = κ } } ρ ν i : ν lim(s α ) and ρ ν i X α ν, for some ν i ν with ot(c ν ) i. Then Y α E α =, because if ρ ν i Y α, then ρ ν i C ν, since i ot(c ν ) and ot(c ν ) < α ν, when ν i ν. Let U ν : ν S 1 be a sequence witnessing that ours is a universal morass. Then U ν ν ν is such that if ν S ατ τ, then U ν = U τ (ν ν); U = ν S U ν is such that U ν} : κ < ν < κ + enumerates all bounded subsets of κ + ; and if ν i ν, then π νν : ( ν,, U ν ) Σ0 (ν,, U ν ). We are now prepared to finish the definition of X α. To be explicit, let us say that W ν codes C α and τ i : i < α if W = <0, > : C } <i + 1, τ i > : i < α }. If there exists ν S α and a δ < ν such that U ν δ} codes C and τ i : i < α, where C is club in α ν and has order type α ν ; τ i E α ν; F α (τ i, τ j ) = 1, for all i < j < α; and γ C i<γ ρ τ i γ / Y α ; then let ν be the least such element of S α and let δ be the least such element of ν and set X α = Y α τ i : i < α }. Otherwise, set X α = Y α. Claim. If ν i ν and i ot(c ν ), then ρ ν i i ρν i. Proof: Set ρ = ρ ν i and ρ = ρ ν i. By tree coherence, it can be seen that ρ = sup( π νν ρ ). Set τ = π νν ( ρ). Then π ρτ = π νν ( ρ + 1). So ρ = sup ( π ρτ ρ ). Hence ρ i ρ. 18

19 5. INTRACTABLE SETS IN L Claim. Suppose that ν i ν and that i ot(c ν ). Then ρ ν i X αν iff ρ ν i X α ν. Proof: ( ) is immediate by the definition of Y αν. To prove ( ), proceed by induction on α ν. Suppose that ρ ν i X α ν. Say ρ ν i = ρτ j, where ρ τ j X α τ, for some τ i τ with ot(c τ ) j. Note that i = j by level coherence. By the previous claim, we have that ρ ν i i ρ ν i and ρ τ i i ρ τ i, so ρ ν i and ρ τ i are j - comparable, since ρ τ i = ρν i. If ρ τ i j ρ ν i, then ρ ν i X α ν, using the definition of Y α ν. On the other hand, if ρ ν i i ρ τ i, then ρ ν i X, by induction, using that ρ τ α ν i X. α τ If τ κ + has cofinality κ, set P τ = i < κ : ρ τ i X κ }. Note that in this case ptn(x, τ) is P τ, modulo the non-stationary ideal on κ. Claim. Suppose that cf(τ) = cf(ν) = κ and that F (τ, ν) = 0. Then P τ P ν is non-stationary. Proof: Say τ < ν. Note that C = α S 0 : ν S α τ S α ν ( ν i ν and π νν ( τ) = τ and ot(c τ ) = ot(c ν ) = α ) } is club in κ. Fix α C and suppose that α P τ P ν, for the sake of a contradiction. Then ρ τ α, ρ ν α X κ. Hence ρ τ α, ρ ν α X α. Furthermore, τ = ρ τ α, since ot(c τ ) = α. Similarly ν = ρ ν α. Hence τ, ν X α. And τ, ν E α, so τ, ν / Y α. It follows that F α ( τ, ν) = 1. But π νν ( τ) = τ and π νν ( ν) = ν. Hence F κ (τ, ν) = 1, rather than 0. Claim. Suppose that τ i : i < κ } is F -homogeneous to 1. Then i<κ P τi is stationary in κ. Proof: Suppose not. Let ν S κ and δ < ν be least such that U ν δ} codes sets C and τ i : i < κ, where τ i : i < κ } is F -homogeneous to 1 and C is club in κ and disjoint from i<κ P τi. Note first that, for fixed i < κ, club many α < κ satisfy (C1) ν S α τ S α ν ( ) ν i ν & π νν ( τ) = τ i & ot(c τ ) = α. Then α that lie in the diagonal intersection of club sets as in (C1) satisfy (C2) ν S α i<α τ i S α ν ( ) ν i ν & π νν ( τ i ) = τ i & ot(c τi ) = α. Suppose that α S 0 and ν S α. Say that U ν provides a counterexample if, for some δ < ν, U ν δ} codes D and η i : i < α, where D is club in α, where η i E α ν, where D i<α Pη α i =, and where Pη α = i < α : ρ η i X α }. Using our 19

20 5. INTRACTABLE SETS IN L minimal choice of ν, note that the set of α satisfying the following condition is club in κ: ( (C3) ν S α ν i ν & ν S α ν ( U ν does not provide a counterexample )) Fix α lying in C and in the club sets described in (C2) and (C3). Then at stage α we set X α = Y α τ i : i < α }, where ν S α and ν i ν and π νν ( τ i ) = τ i, for all i < α. Consequently ρ τ i α = τ i X α, for all i < α. Thus ρ τ i α X κ, for all i < α. Therefore α i<κ P τi. But α C. 6. The characterization problems The characterization problems are not first-order statements. In this section (especially) let V indicate a countable standard transitive model of ZFC and, having fixed V, let L denote the constructible sets in the sense of V. We shall sketch proofs of three examples. In outline, we shall see that (1) none of the characterization problems is solvable in the minimum model; (2) if V is sufficiently non-minimal, the characterization problems are solvable in V ; however, (3) V has a minimalized outer model in which none of the characterization problems is solvable. For the sake of clarity, we have resisted the temptation to state the strongest possible results the proofs support. This is why the results in this section are dubbed examples. Example 1. In the minimum model of ZFC, the tree and partition problems are unsolvable at all κ + and the stationary set characterization problem is unsolvable at κ + when κ ω 1 is regular. Proof: Let L min be the minimum model of ZFC, that is, L min is L α, where α is least such that L α ZFC (assuming such α exist). We shall exhibit an L-definable operation taking parameter-free sentences ϕ in the language of set theory to trees T ϕ of height and cardinality κ + such that T ϕ has a cofinal branch in a κ + -preserving outer model iff L ϕ. This suffices to see that the tree characterization problem is unsolvable because ϕ : L min ϕ } / L min. Indeed, a structure isomorphic to L min is definable from ϕ : L min ϕ }. Declare that t T ϕ iff t: [κ, α) 2, for some α < κ + and, if α is such that L β ZF 2 + = κ +, then either L β [t ] is not admissible or L β [t ] ν : <3, ν> Decode(t ) } is a club class, and if <3, ν> Decode(t ), then ϕ L ν. T is ordered by functional extension. On the one hand, if T ϕ has a cofinal branch b in a κ + -preserving outer model, then Decode(b) provides a club class of ordinals ν such that ϕ L ν. It follows that L ϕ. Conversely, if L ϕ then there exists an L-definable club class of ordinals ν such that ϕ L ν. Using this it is straightforward to modify the proof of the Tree Theorem to produce an L-definable class forcing P ϕ such that forcing with P ϕ preserves all L-cardinals and the GCH and adds a cofinal branch to T ϕ. 20